I'll try to bend David White's answer towards the actual situation of your question. The outcome is somewhat clumsy and it totally looks like model structures can be eliminated from it, but anyway:
Assume your category C is closed monoidal and locally presentable. Then it is a monoidal model category with cofibrations and fibrations all morphisms and weak equivalences the isomorphisms. This model category is cofibrantly generated: One can take the identity of the initial object as generating trivial cofibration and the set of all morphisms between the objects $G$ from some generating set as generating cofibrations.
This model category then satisfies the hypotheses of Barwick's Thm. 4.46 in this article for a Bousfield localization at the one element set containing $f$. The homotopy category for the localized model structure has the universal property you want and lives in the same universe. You have an adjunction between the homotopy category of the original model structure, which is the category itself, and the localization.
This adjunction is a reflection to an orthogonal subcategory as in Adamek/Rosicky, 1.35-1.38, namely to the full subcategory of all objects from whose point of view $f$ "was already an isomorphism" (i.e. $f$-orthogonal objects; precise definition via a unique-lifting-condition). This is much like in your example (but with the condition on the twist removed and without the domain of f having to be special). If you chase through the proofs, you also get an expression of the reflection functor as a colimit via the small object argument, resp. via Adamek/Rosicky's proof...
Barwick's Prop. 4.47 gives then a criterion for the homotopy category to be closed monoidal again: It suffices that any object $X$ which satisfies the unique right lifting condition with respect to $f$ also satisfies it with respect to $f \otimes G$ for every generating object $G$ (remember the category was locally presentable now) i.e. if $f$ induces an iso $Hom(f,X)$ then $Hom(f \otimes G,X)$ is an iso, too, for every generating object $G$.
edit: Sorry, I am no longer sure that the homotopy category of the Bousfield localization is in fact the localization along $f$ in the sense you asked for: When you localize with respect to an arrow you automatically invert together with it a bunch of other arrows. When you do plain category theory it is somewhat uncontrollable what those other arrows are, it seems to me. When you do Bousfield localization these other arrows are those having the left lifting property with respect to the $f$-local objects. Now I don't see a reason why the class of additionally inverted arrows should be the same in both cases. What Bousfield localization as sketched here probably yields, is the universal colimit preserving functor which inverts $f$.
My short answer, is that in the great majority of situation where this $\mathcal{Y}/f^*$ plays a role, it is a mistake to see it as a topos. There are exceptions, but most of the time it is not meant to be a topos, but rather a $\mathcal{X}$-indexed topos.
To clarify the discussion, I will follow Joyal's terminology:
I'm calling "Logos" what we usually call a Grothendieck topos. Morphisms of logos are the continuous left exact functor, i.e. the $f^*$.
Toposes are the object of the opposite category of the category of logos. If $\mathcal{X}$ is a topos I denote by $Sh(\mathcal{X})$ the corresponding logos, which I think of as the category of sheaves of sets over $\mathcal{X}$. If I say "$x \in \mathcal{X}$" I mean that $x$ is a point (or maybe a generalized point) of $\mathcal{X}$.
This is mean't to mimic the picture of the conection between frames and locales (where the frame corresponding to a locales is denoted by $\mathcal{O}(\mathcal{X})$.
Now, The topos corresponding to the logos $Sh(\mathcal{Y})/f^*$ is the following object:
$$ \left(\mathbb{S} \times \mathcal{Y} \right) \coprod_{\mathcal{Y}} \mathcal{X}$$
where $\mathbb{S}$ is the Sierpinski topos, i.e. corresponding to the topological space with one closed point and one open point. (this is easily seen using that colimits of toposes corresponds to limits of logos, which are justs limits of categories)
So you can think of it as a kind of "cone construction" where the Sierpinski space is used as an interval. This object can indeed be interesting, and you can somehow guess that it will be especially interesting for the theory of local morphisms as you mentioned in your post.
But, in my opinion, this topos as simply nothing to do with the idea of working with $\mathcal{Y}$ as "an object over $\mathcal{X}$", i.e. to try to think of the map $\mathcal{Y} \rightarrow \mathcal{X}$ by somehow looking the fiber $\mathcal{Y}_x$ for $x \in \mathcal{X}$ and how it varies when $x$ varies in $\mathcal{X}$. Which is what we want to do in all the case you mentioned:
A proper map is a map whose fiber are compacts, in a "nicely locally uniforme way""
A locally connected map is map whose fiber are locally connected in a nicely locally uniform way.
A separated map... well you can actually also see it as a map whose fiber are separated in a locally uniforme way, but that is not quite what the definition you gave really says. The way I think about this definition is that when you have a class of map stable under composition and pullback then it is nice to consider the class of maps whose diagonals have this property, because you then get for free the lemma that "if $f \circ g$ is proper and $f$ is separated then $g$ is proper".
Let's now look at what happen when you want to work with $\mathcal{Y}$ as an object over $\mathcal{X}$ :
Essentially, instead of looking at $Sh(\mathcal{Y})$ you want to look at something like $Sh(\mathcal{Y}_x)$ for all $x \in \mathcal{X}$, which should give you some kind of family of logoses parametrized by $x \in \mathcal{X}$.
In the same way that the correct notion of "familly of set parametrized by $x \in \mathcal{X}$ is a sheaves over $\mathcal{X}$ the correct notion of such familly of logos is something like "a sheaves of logos". It is not quite just a "sheaves in the category of logos" because the definition of logos involved some infinitary operations (the infinite coproduct/colimits) which you want to replace by $Sh(\mathcal{X})$-indexed colimits/coproducts. So the correct notion is what I would call an "internal logos" and is a special kind of sheaf of logos (It is exactly a sheaf of logos which admits $Sh(\mathcal{X})$-indexed disjoint and universal coproducts).
Also note that these sheaf of logos in particular gives you the type of structure that you asked about in your last question. (they satisfies stronger property though, like Beck Cheavely conditions related to the fact that they have indexed colimits)
Then for technical reason, we tend to look at sheaves of categories rather as indexed categories or fibered categories, that is why you end up with a fibration over $Sh(\mathcal{X})$.
But if you somehow forget that it was mean't to be a "sheaf of logos" over $Sh(\mathcal{X})$ and see the total category of the fibration as a new logos, then you just get a completely different and new object that have very little to do with what I was describing.
I'm finishing with an informal discussion of why the $\mathcal{X}$-indexed logos corresponding to $\mathcal{Y} \rightarrow \mathcal{X}$ should be $Sh(\mathcal{Y})/f^*$. This is not a proof, just some kind of heuristic arguement. The proof is justs that there is a relatively deep theorem saying that the category of $Sh(\mathcal{X})$-indexed logoses is equivalent to the category of toposes over $\mathcal{X}$ and that the equivalence is given by this construction, and is compatible with pullbacks along maps $\mathcal{X'} \rightarrow \mathcal{X}$.
So What should be the "sheaf of logos" (or internal logos) corresponding to $\mathcal{Y} \rightarrow \mathcal{X}$. I need to think about "what should be its section over some étale cover $p: \mathcal{E} \rightarrow \mathcal{X}$".
I want something that, in a continuous way associate to each $e \in \mathcal{E}$ a sheaves over $\mathcal{Y}_p(e)$, i.e. something that continuously in $e \in \mathcal{E}$ and $y \in \mathcal{Y}_{p(e)}$ associate a set. So basically it is a sheaf of set over $\mathcal{Y} \times_{\mathcal{X}} \mathcal{E}$.
Now if $\mathcal{E}$ is the etale space of a sheaves $ E \in Sh(\mathcal{X})$, one has that $Sh(\mathcal{E}) = Sh(\mathcal{X})/E$, and $Sh(\mathcal{Y} \times_{\mathcal{X}} \mathcal{E}) = Sh(\mathcal{Y})/f^* E$
So in the end you do get the fibered category we are talking about.
Best Answer
I'm not an expert in differential geometry, but it seems to me that the property of the Liouville $1$-form only talks about the given manifold and its tangent bundle, no other manifolds are involved, hence this isn't a universal property. But I think that we can generalize the property as follows:
Let $\mathsf{Mfd}/X$ denote the category of smooth manifolds $Y$ equipped with a smooth map $Y \to X$. Consider the functor $(\mathsf{Mfd}/X)^{\mathrm{op}} \to \mathsf{Vect}$ which maps $Y \to X$ to $\Omega^1(Y)$. Then I claim that this functor is represented by the Liouville $1$-form $(T^* X \to X,\lambda_X)$. This means: Given $Y \to X$ and $\omega \in \Omega^1(Y)$, there is a unique smooth map $f : Y \to T^* X$ over $X$ such that $f^* \lambda_X = \omega$.
In fact, one defines $f$ to be the composition $Y \xrightarrow{\omega} T^* Y \to T^* X$. Then $f^* \lambda_X = \omega^* \lambda_Y = \omega$.
EDIT: As mentioned in the comments, one has to take relative differential forms $\Omega^1(Y/X)$.