Besides that the theories (étale fundamental group and Tannakian formalism) just formally look alike, there exist actual comparison results between certain étale and Tannakian fundamental groups.
Namely: there is Nori's fundamental group scheme $\pi_1^N(S,s)$, where $S$ is a proper and integral scheme over a field $k$ having a $k$--rational point. It is defined to be the fundamental group of some Tannakian category associated with $S$ (to be precise: The full $\otimes$-subcategory of the category of locally free sheaves on $S$ spanned by the essentially finite sheaves). In the case $k$ is algebraically closed, there is a canonical comparison morphism from Nori's fundamental group to Grothendieck's étale fundamental group, and if moreover $k$ is of characteristic zero this morphism is an isomorphism.
One more comment: The classical Tannaka-Krein duality theorem for compact topological groups (see e.g. Hewitt&Ross vol. II) should presumably be another realisation of the common generalisation you seek.
The basic Grothendieck's assumptions means we are dealing with an connected atomic site $\mathcal{C}$ with a point, whose inverse image is the fiber functor $F: \mathcal{C} \to \mathcal{S}et$:
(i) Every arrow $X \to Y$ in $\mathcal{C}$ is an strict epimorphism.
(ii) For every $X \in \mathcal{C}$ $F(X) \neq \emptyset$.
(iii) $F$ preseves strict epimorphisms.
(iv) The diagram of $F$, $\Gamma_F$ is a cofiltered category.
Let $G = Aut(F)$ be the localic group of automorphisms of $F$.
Let $F: \widetilde{\mathcal{C}} \to \mathcal{S}et$ the pointed atomic topos of sheaves for the canonical topology on $\mathcal{C}$. We can assume that $\mathcal{C}$ are the connected objects of $\widetilde{\mathcal{C}}$.
(i) means that the objects are connected, (ii) means that the topos is connected, (iii) that $F$ is continous, and (iv) that it is flat.
By considering stonger finite limit preserving conditions (iv) on $F$ (corresponding to stronger cofiltering conditions on $\Gamma_F$) we obtain different Grothendieck-Galois situations (for details and full proofs see [1]):
S1) F preserves all inverse limits in $\widetilde{\mathcal{C}}$ of objets in $\mathcal{C}$, that is $F$ is essential. In this case $\Gamma_F$ has an initial object $(a,A)$ (we have a "universal covering"), $F$ is representable, $a: [A, -] \cong F$, and $G = Aut(A)^{op}$ is a discrete group.
S2) F preserves arbritrary products in $\widetilde{\mathcal{C}}$ of a same $X \in \mathcal{C}$ (we introduce the name "proessential for such a point [1]). In this case there exists galois closures (which is a cofiltering-type property of $\Gamma_F)$, and $G$ is a prodiscrete localic group, inverse limit in the category of localic groups of the discrete groups $Aut(A)^{op}$, $A$ running over all the galois objects in $\mathcal{C}$.
S2-finite) F takes values on finite sets. This is the original situation in SGA1. In this case the condition "F preserves finite products in $\widetilde{\mathcal{C}}$ of a same $X \in \mathcal{C}$ holds automatically by condition (iv) ($F$ preserves finite limits), thus there exists galois closures, the groups
$Aut(A)^{op}$ are finite, and $G$ is a profinite group, inverse limit in the category of topological groups of the finite groups $Aut(A)^{op}$.
NOTE. The projections of a inverse limit of finite groups are surjective. This is a key property. The projection of a inverse limit of groups are not necessarily surjective, but if the limit is taken in the category of localic groups, they are indeed surjective (proved by Joyal-Tierney). This is the reason we have to take a localic group in 2). Grothendieck follows an equivalent approach in SGA4 by taking the limit in the category of Pro-groups.
S3) No condition on $F$ other than preservation of finite limits (iv). This is the case of a general pointed atomic topos. The development of this case we call "Localic galois theory" see [2], its fundamental theorem first proved by Joyal-Tierney.
[1] "On the representation theory of Galois and Atomic topoi", JPAA 186 (2004)
[2] "Localic galois theory", Advances in mathematics", 175/1 (2003).
Best Answer
The point of view where this title comes from is that Grothendieck's theorem can be seen as a characterization of toposes of the form $BG$ for $G$ a profinite group. It shows that some toposes can be represented as $BG$.
I think before Joyal–Tierney's paper it was also known how to generalize from profinite group to general localic groups.
Joyal–Tierney's theorem shows that if you replace "pro-finite group" by "localic groupoid" then you actually get all Grothendieck toposes this way.
You can't directly recover Grothendieck's theorem from Joyal–Tierney's theorem in the sense that the theorem as stated above doesn't tell you for which toposes the localic groupoid can be chosen to be a profinite group. But if you are familiar with the method used in the paper and how the groupoid is obtained (which in my opinion are even more important than the theorem itself) then it is fairly easy to recover Grothendieck's theorem. For example, it immediately follows from Joyal–Tierney's paper that a topos is of the form $BG$ for $G$ a localic groups if and only if it admits a point $* \to \mathcal{T}$ which is an open surjection (which does feel similar to Grothendieck's theorem in terms of a fiber functor).
Regarding the use of internal logic, it is definitely not essential, it just makes everything simpler (at least if you are ok with its use) but one could do without it.
The main way they use internal logic is that in the first sections they prove some results about sup-lattice and frames locales, that are later applied not to sup-lattices and frames, but to sup-lattices and frames in a topos $\mathcal{T}$. I believe there are also a few places where they make a claim about a morphism of locales $f:X \to Y$ and then only prove it when $Y$ is the point (sorry I don't have the paper with me to give precise reference).
Another place where one can consider they use a bit of internal logic — though this one might be only at the level of intuition — is when they show that every topos admits an open cover by a locale. They do this by considering the topos as a classifying topos of some theory $T$ and considering the propositional theory $T'$ of "enumerated $T$-models", that is, $T$-models that are explicitly given as a subquotient of the natural numbers. Though if I remember correctly, they present the argument in a way that doesn't directly involve any logic… (and in any case there are other proofs of this results that are purely in terms of sites, for example the one in MacLane and Moerdijk's book "Sheaves in geometry and logic").