Let me try to cut through the jargon. One thing that confuses me are two uses of "tangent spaces" here, that I believe are quite unrelated. One is the usual notion of tangent spaces of smooth manifolds say, based on which one defines differentials, and all sorts of de Rham cohomology etc.; I believe the "differential cohomology" is of this sort. On the other hand, there is the notion of the "tangent $\infty$-topos". I vaguely understand the reason for also calling this "tangent", but it is by a series of analogies, and there seems to be no relation between these two concepts in the case at hand.
More specifically, for condensed anima, the tangent $\infty$-topos is simply the $\infty$-category of pairs $(X,A)$ where $X$ is a condensed anima and $A\in \mathcal D(\mathrm{Cond}_{/X},\mathbb S)$ is a hypercomplete sheaf of spectra on the site of condensed sets over $X$. This is definitely a very interesting structure. This whole concept of $6$-functor formalisms is very much about such categories. Working with torsion coefficients, and allowing only "relatively discrete" coefficients, I've developed something along those lines in Etale cohomology of diamonds. Something even closer is in Chapter VII of Geometrization of the local Langlands correspondence (link should be active in a few days is active), where we restrict to the solid objects in $\mathcal D(\mathrm{Cond}_{/X},\mathbb Z_\ell)$. A critical role is then played by the left adjoint $f_\natural$ to pullback $f^\ast$. These do not exist in any classical setup, but have excellent formal properties. In fact, one gets a variant of a $6$-functor formalism where homology and cohomology are now on equal footing again (and arguably homology is even more primitive, again): The pullback functor $f^\ast$ admits a left adjoint $f_\natural$ ("homology") and a right adjoint $Rf_\ast$ ("cohomology"), both of which commute with any pullback. Moreover, Poincare duality holds for proper smooth maps. So yes, there's something interesting about this.
On the other hand, the reference to the differential cohomology hexagon in the question confuses me. For condensed sets, there are no tangent spaces, no differential forms, etc., and you can't get them back by magic. I think the problem is that I have no idea what the term "differential cohesion" means, but my strong feeling is that to have "differential cohesion" one needs extra structure like tangent spaces on the model spaces (and that passing to the "tangent $\infty$-topos" is not at all supplying these, as this is a very different procedure).
$\require{AMScd}$
Edit: Thanks to the David's for their enlightening comments! Now I understand that the part with the differentials is really only in the examples, not in the "differential cohomology hexagon". For my convenience, let me reformulate this hexagon in my own language.
Say $X$ is a condensed anima, and $A\in \mathcal D(\mathrm{Cond}_{/X},\mathbb S)$ is a sheaf of spectra on condensed sets over $X$. (Or take $X$ a scheme, and $A\in \mathcal D(X_{\mathrm{proét}},\mathbb Z_\ell)$, or $X$ a small v-stack and $A\in \mathcal D(X_v,\mathbb Z_\ell)$, or...) Let $\pi$ denote the projection from the site of $X$ to the (pro-étale) site of the point. Then pullback $\pi^\ast$ has a right adjoint $R\pi_\ast$ ("cohomology") and a left adjoint that I will denote $\pi_\natural$ ("homology").
In particular, we get condensed spectra $R\pi_\ast A$ (=$R\Gamma(X,A)$), the cohomology of $A$, and $\pi_\natural A$, the homology of $A$. By adjunction, we get maps
$$
\pi^\ast R\pi_\ast A\to A\to \pi^\ast \pi_\natural A.
$$
Let $\overline{A}=\mathrm{cofib}(\pi^\ast R\pi_\ast A\to A)$ and $\tilde{A}=\mathrm{fib}(A\to \pi^\ast \pi_\natural A)$. Then there is a pullback square
$$\begin{CD}
A @>>> \overline{A}\\@VVV @VVV\\ \pi^\ast \pi_\natural A @>>> \pi^\ast \pi_\natural \overline{A}
\end{CD}
$$
and a pushout square
$$\begin{CD}
\pi^\ast R\pi_\ast \tilde{A} @>>> \pi^\ast R\pi_\ast A\\@VVV @VVV\\ \tilde{A} @>>> A
\end{CD}
$$
Of course, pushout squares and pullback squares are equivalent, but I want to stress that one wants to use them to recover $A$ from "simpler" information. However, to me $\tilde{A}\to A\to \overline{A}$ all feel extremely similar, and I'd regard these squares as simple statements about how to analyze the small difference between them.
I think there are a few things to untangle here.
First, as concerns your highlighted question, it seems that you've answered it yourself: outside the compact Hausdorff case (where the uniform structure is completely equivalent to the topology), it's unreasonable to think that the associated condensed set "knows" the uniform structure. It really only knows the topology.
Second, you are correct in your unpacking of the definition of "solid" as it applies to (let's say metrizable) toplogical abelian groups $M$. If $\underline{M}$ is solid, then for any nullsequence $(m_n)$ in $M$ and any sequence of integers $(a_n)$ the sum $\sum a_nm_n$ must converge in $M$. I agree that this feels a lot saying that $M$ is nonarchimedean and complete. However, though I gather for you "complete" means "Cauchy-complete" as usual, I'm not sure what general definition of a topological abelian group $M$ being "nonarchimedean" you're thinking of here to say that you think it is actually equivalent.
Here's something I find helpful to keep in mind. There are several differences between the notion of "solid" and the notion of "complete", beyond the fact that solid enforces some kind of non-archimedeanness. (Indeed, the same remarks apply in the liquid setting, which does not enforce nonarchimedeanness.)
First, while both the definition of "complete" and the definition of "solid" are of the form "for every [...] there exists a unique {...}", the differences in nature of the ...'s occuring lead to drastic divergence of the general notions. Most importantly, the solid condition in no way implies any kind of Hausdorff behavior. Essentially, the reason is that in the solid condition you already require in [...] a bunch of limits to exist uniquely (because you map in from a compact Hausdorff space). Then the solid condition is only that this generates more limits (and uniquely) by taking certain $\mathbb{Z}$-linear combinations. Whereas in the usual Cauchy completeness every limit that you posit exists, you also posit exists uniquely.
This phenomenon, that solid abelian groups incorporate non-Hausdorff behvaior, it absolutely crucial to having a functioning theory. Why? Because non-Hausdorff behavior is inevitable once you require an abelian category. If $M$ is some non-discrete condensed abelian group which you want to call "complete", and $N$ is any discrete dense subgroup mapping into $M$, the cokernel has to be "complete" as well, even though it's a classic example of a "bad quotient" which is non-Hausdorff.
(This is an example of the trade-off between "good categories of (mostly) bad objects" vs. "bad categories of good objects". But for me personally, I've seen enough examples of the solid formalism gracefully handling non-Hausdorff spaces in practice that I no longer think of them as "bad objects".)
You might be tempted to then think that solid abelian groups are more like weakened analog of complete topological abelian groups, where you drop the uniqueness requirement in the definition of completeness and only require existence of limits for Cauchy sequences. But no, the fact that solidness is an "exists a unique" condition is also crucial for it being an abelian category. "Exists" is just not stable enough a notion.
The other big difference is that a solid abelian group is "only required to be complete as far as compact subsets are concerned". If you have a Cauchy sequence which is not contained in a compact subset, the solid condition says nothing about it. Thus, for example, solid abelian groups are closed under arbitrary direct sums, whereas I don't think there's any reasonable topology on the direct sum $\oplus \mathbb{Z}_p$ for which it's complete [Edit: I thought wrong! See https://mathoverflow.net/questions/387322/countable-sum-bigoplus-n-0-infty-mathbb-z-p-as-a-topological-group]. Again, the fact that solid abelian groups are closed under all colimits is very important for us theoretically: it's part of what makes sure that the category of solid abelian groups "behaves like the category of modules over a ring" (formally, it is an abelian category generated by compact projective objects), and therefore has convenient algebraic properties.
Now, it seems the main thrust of your question as about defining possibile non-abelian analogs of solidness. I don't want to say this can't be done (I doubt it can but I certainly could be wrong), but I hope that the above remarks show that if you want to define such a notion, you shouldn't do it by trying to follow the usual presentation via Cauchy-completeness and uniform structures. Despite the fact that the two notions agree on many familiar objects, there is a huge divergence in general.
Best Answer
Thanks for the question! One interpretation of the conjecture is true. Let me elaborate. The following results are kind of implicit in some discussion towards the end of www.math.uni-bonn.de/people/scholze/Analytic.pdf (see especially Proposition 13.16, Propositon 14.7, and some surrounding discussion), although some relevant computations are not explicitly done there, but in the Master Thesis of Grigory Andreychev (I hope it will be public soonish):
To any Huber pair $(A,A^+)$, that is a pair of a certain kind of (always assumed complete here) topological ring $A$ together with an open and integrally closed subring $A^+\subset A$ consisting of powerbounded elements, one can associate an analytic ring $(A,A^+)_{\blacksquare}$. This is Proposition 13.16. Analytic rings are pairs of a condensed ring $B$ together with a notion of "completeness" for condensed $B$-modules: See Lecture 7 of www.math.uni-bonn.de/people/scholze/Condensed.pdf for the "non-animated" version (and Lecture 12 of Analytic.pdf for a more general version). As remarked in Remark 13.17, Andreychev has proved that this has the property that $(A,A^+)_{\blacksquare}[S]$ is concentrated in degree $0$ for any profinite set $S$, so it's an analytic ring in the "non-animated" sense (where everything is a usual condensed ring, and condensed module).
This functor from Huber pairs to analytic rings is fully faithful. Again, this is part of Proposition 13.16. It is in this sense that the analytic geometry defined in these lectures extends the category of adic spaces.
The functor $(A,A^+)\mapsto (A,A^+)_{\blacksquare}$ always has underlying condensed ring the condensed ring $\underline{A}$ corresponding to the topological ring $A$, but the notion of completeness for modules depends on the subring $A^+\subset A$. From this perspective, the seemingly obscure conditions on $A^+$ have a very natural interpretation, see Remark 13.18: Talking about such subrings of $A$ is just one way to talk about the associated analytic ring structures (other subrings of $A$ can also lead to such analytic ring structures, but for all such analytic ring structures there would be a different choice of subring that satisfies the conditions imposed on $A^+$).
The functor $(A,A^+)\mapsto (A,A^+)_{\blacksquare}$ extends to more general pairs of a condensed animated ring $A$ (where "animated" = "simplicial, up to homotopy"), satisfying some conditions, and equipped with a similar subalgebra $A^+\subset \pi_0 A$. (Technically, it is enough if $A$ is nuclear over $\mathbb Z[[X_1,\ldots,X_n]]$ for some $n$.)
For any rational open subset $U\subset \mathrm{Spa}(A,A^+)$, one can naturally define a condensed animated $A$-algebra $\mathcal O_X(U)$ together with a subalgebra $\mathcal O_X^+(U)\subset \pi_0 \mathcal O_X(U)$, fitting into the class implicit in 4). In particular, there is an associated analytic ring $(\mathcal O_X(U),\mathcal O_X^+(U))_{\blacksquare}$. This is the localization of the analytic ring $(A,A^+)_{\blacksquare}$ to the preimage of $U$ in $\mathrm{AnSpec}((A,A^+)_{\blacksquare})$ via the map of Proposition 14.7.
The association $U\mapsto \mathcal O_X(U)$ defines a sheaf of condensed animated $A$-algebras on $\mathrm{Spa}(A,A^+)$, for any Huber pair $(A,A^+)$. This is a special case of Proposition 12.18.
If $(A,A^+)$ satisfies any of the classical criteria of being sheafy, or generally if $A$ is Tate and sheafy, then $\mathcal O_X(U)$ is concentrated in degree $0$ and comes from the usual structure sheaf of Huber rings. This is Proposition 14.7. (The general Tate case uses some results of Kedlaya. It may be that by recent work of Ramero http://math.univ-lille1.fr/~ramero/CoursAG.pdf, Chapter 12, who extends some of Kedlaya's work to the case of general Huber rings, simply asking $(A,A^+)$ to be sheafy is enough.)
Now conversely, if the sheaf $\mathcal O_X(U)$ of condensed animated $A$-algebras happens to be concentrated in degree $0$, and also to be quasiseparated, then it agrees with the (presheaf of condensed rings associated to the) presheaf of Huber rings defined by Huber, which is thus sheafy.
That final point 8) is, I think, a partial answer to your question. At least if $A$ is Tate, it shows that sheafyness is equivalent to the assertion that for all rational $U\subset \mathrm{Spa}(A,A^+)$, the condensed animated $A$-algebra $\mathcal O_X(U)$ is concentrated in degree $0$ and quasiseparated. One could wonder whether the "... and quasiseparated" ending is necessary; my gut feeling is that it is necessary.
Upshot: Huber was working in the context of complete topological rings (satisfying some conditions), and had to insist that his structure presheaf stays in the same realm. If you allow yourself more flexibility, in particular a notion of completeness that does not entail separatedness, and a formalism of topological rings that allows for higher homotopy groups (i.e., animated rings), then one can define a better version of his presheaf, that is always a sheaf. (See the beginning of Lecture 11 in Analytic.pdf, especially page 73, for some intuitive explanations.) Then sheafyness of Huber pairs is simply the question whether this more general construction stays in the classical realm, which fortunately happens so far in all cases of interest. I should mention that Bambozzi-Kremnizer have recently reached similar results, using different foundations, see arXiv:2009.13926. (In their approach, the role of $A^+$ is less clear.)
Edit: Let me actually be much more explicit about all of this. Consider a rational subset $U\subset \mathrm{Spa}(A,A^+)$, explicitly given as the locus $U=\{|f_1|,\ldots,|f_n|\leq |g|\neq 0\}$, for $f_1,\ldots,f_n,g\in A$ generating an open ideal. In that case, Huber's ring $\mathcal O_X^H(U)$ is given by $$ A\langle T_1,\ldots,T_n\rangle[\tfrac 1g]/\overline{(f_1-gT_1,\ldots,f_n-gT_n)}. $$ Intuitively, this is just expressing that on this subset $g$ is invertible, and $T_i=\tfrac{f_i}g$ has absolute value $\leq 1$, so we can allow all convergent power series in the $T_i$. To stay in the setting of complete topological rings, it is necessary to take the quotient by the closure of this ideal.
On the other hand, a theorem of Kedlaya (https://kskedlaya.org/papers/aws-notes.pdf, Theorem 1.2.7) shows that if $\mathcal O_X^H$ is a sheaf and $A$ is Tate, then it is not necessary to take the closure. Moreover, his results show that the sequence $(f_1-gT_1,\ldots,f_n-gT_n)$ is (Koszul-)regular. In other words, in this case $\mathcal O_X^H(U)$ is computed by the Koszul complex $$ A\langle T_1,\ldots,T_n\rangle[\tfrac 1g]/^{\mathbb L}(f_1-gT_1,\ldots,f_n-gT_n), $$ where I write, for an $A$-module $M$ and elements $a_1,\ldots,a_n\in A$, $$ M/^{\mathbb L}(a_1,\ldots,a_n) $$ for the (homological) Koszul complex $$ 0\to M\xrightarrow{(a_1,\ldots,a_n)} M^n\to \ldots\to M^n\xrightarrow{(a_1,\ldots,a_n)} M\to 0. $$
What condensed mathematics allows you to do is to consider the derived quotient $$ A\langle T_1,\ldots,T_n\rangle[\tfrac 1g]/^{\mathbb L}(f_1-gT_1,\ldots,f_n-gT_n) $$ as the correct answer in general. For this, you have to consider it simultaneously as endowed with (something like) a topology, as a complex, and as a (commutative) algebra. Thus, you have to mix higher (i.e. homotopical) algebra with topology, and this is what condensed mathematics can easily accomodate. In fact, these are just the condensed animated rings I've been talking about above. (Animated rings are, at least in characteristic 0, just the "commutative algebras in the derived category", correctly understood; making them condensed amounts to putting something like a topology on them, in particular on their homotopy groups.)
In particular, what my answer above means is the following, at least if $A$ is Tate: $\mathcal O_X^H$ is a sheaf if and only if for all $U$ as above, the sequence $(f_1-gT_1,\ldots,f_n-gT_n)$ is Koszul-regular and generates a closed ideal. This is in fact already a theorem of Kedlaya, I believe.