If theory (in some logic) has 2 different models, are these models related in some other way? Let us assume, that we work in sound and complete logic. Let us assume that we have theory Th and 2 different models Mod1 and Mod2 that are interpretations of this theory Th. My question is - what we can say about how Mod1 is related to Mod2? E.g. one can imagine the interpretation of Th into topological spaces (it is possible for some logics), what can we say about the relation betwen the algebraic-topology invariants of those 2 models? Or is there any other, more modest results about relation between the models who are interpretations of the same theory?
 A: I suspect you're thinking of complete theories; or, to put it another way, you're looking for a structural characterization of the relation of (first-order) elementary equivalence. Incomplete theories don't generally say much. For example, any two structures trivially satisfy the empty theory, and any two groups - however different - are models of the theory of groups.

Main answer
For simplicity let's restrict attention to structures in relational languages. In this case, there is in fact a good structural relation we can conclude, namely the existence of an appropriate "back-and-forth family."
We start with the notion of a partial isomorphism:

If $\mathfrak{A},\mathfrak{B}$ are structures in the same language, a partial isomorphism from $\mathfrak{A}$ to $\mathfrak{B}$ is an isomorphism $p:\mathfrak{A}'\cong\mathfrak{B}'$ with $\mathfrak{A}'\subseteq\mathfrak{A}$ and $\mathfrak{B}'\subseteq\mathfrak{B}$. We allow structures to be empty here, so that the empty map is always a partial isomorphism between any two structures.

An individual partial isomorphism is basically useless. What we care about are sets of partial isomorphisms with intereting "gluing" properties:

Given structures $\mathfrak{A},\mathfrak{B}$ in the same language and a linear order $L$, an $L$-back-and-forth-system for $\mathfrak{A}$ and $\mathfrak{B}$ is an indexed family $(P_l)_{l\in L}$ such that:

*

*each $P_l$ is a nonempty set of partial isomorphisms from $\mathfrak{A}$ to $\mathfrak{B}$;


*whenever $l<_Lm$, $p\in P_m$, and $a\in \mathfrak{A}$, there is a $q\in P_l$ with $q\supseteq p$ and $a\in dom(q)$; and


*whenever $l<_Lm$, $p\in P_m$, and $b\in \mathfrak{B}$, there is a $q\in P_l$ with $q\supseteq p$ and $b\in ran(q)$.

(We can actually generalize this a fair bit, e.g. allow $L$ to be a partial order, but that's not super important right now.) Note that $L$-back-and-forth-systems, while not genuine functions, are still "function-like" in many ways; in particular, they can be composed in an appropriate way, and the relation "there is an $L$-back-and-forth system between" is an equivalence relation.
The two most important choices of $L$ here are $\mathbb{N}$ and $\mathbb{N}^*$ (= $\mathbb{N}$ with the opposite ordering). If there is an $\mathbb{N}^*$-back-and-forth system between $\mathfrak{A}$ and $\mathfrak{B}$, we say $\mathfrak{A}$ and $\mathfrak{B}$ are potentially isomorphic (and write "$\mathfrak{A}\cong_p\mathfrak{B}$"). It turns out that this is a very useful notion:

(Karp/Barwise) The following are equivalent for structures $\mathfrak{A},\mathfrak{B}$ in the same language:

*

*$\mathfrak{A}\cong_p\mathfrak{B}$.


*$\mathfrak{A}$ and $\mathfrak{B}$ satisfy the same sentences in the infinitary logic $\mathcal{L}_{\infty,\omega}$.


*There is a forcing extension of the set-theoretic universe in which $\mathfrak{A}\cong\mathfrak{B}$.

Unfortunately, this overshoots our goal. First-order elementary equivalence corresponds to the usual ordering on $\mathbb{N}$:

(Fraisse) The following are equivalent for structures $\mathfrak{A},\mathfrak{B}$ in the same language:

*

*$\mathfrak{A}\equiv\mathfrak{B}$ (= $\mathfrak{A}$ and $\mathfrak{B}$ satisfy the same first-order sentences).


*There is an $\mathbb{N}$-back-and-forth system between $\mathfrak{A}$ and $\mathfrak{B}$.

(We can also talk - more intuitively, but less efficiently - in terms of games and strategies, the relevant term here being Ehrenfeucht-Fraisse games.) Besides being useful for analyzing specific structures and theories, Fraisse's theorem also is a key part of characterizing first-order logic as a whole system - see Lindstrom's theorem.
EF-games, and variations on the same (such as pebble games), are important in finite model theory; see Libkin, Elements of finite model theory.

Coda
Resisting the temptation to leave well enough alone, let me belatedly take this opportunity to say a bit about other logics' elementary equivalence analogue characterizations as well.
Let's start with second-order logic. In a sense, $\equiv_{\mathsf{SOL}}$ is reducible to $\equiv$ + powerset: every structure $\mathcal{M}$ has an associated "power structure" $\mathcal{M}^+$ with underlying set $\mathcal{M}\sqcup\mathcal{P}(\mathcal{M})$, such that $$\mathcal{M}\equiv_\mathsf{SOL}\mathcal{N}\quad\iff\mathcal{M}^+\equiv\mathcal{N}^+.$$ This can be used to tweak the back-and-forth characterization of first-order elementary equivalence to work for second-order logic: basically, we want to look at maps between sets of elements of the structures involved.
On the other hand, the infinitary logic $\mathcal{L}_{\omega_1,\omega}$ - just like $\mathsf{FOL}$ but with countable conjunctions and disjunctions allowed - has a more complicated elementary equivalence relation. The relation $\equiv_{\omega_1,\omega}$ does not correspond to the existence of an $\omega_1$-back-and-forth system; a game-style characterization of $\equiv_{\omega_1,\omega}$ was introduced by Vaanaanen and Wang, but it is more complicated - and in fact it turns out that in a precise sense it is harder for $\mathcal{L}_{\omega_1,\omega}$ to describe its own equivalence relation than it is for $\mathsf{FOL}$ or $\mathsf{SOL}$ (as far as I know this is due to Farmer Schlutzenberg, and extends to other infinitary logics as well).
There is also an important example of a class of logics defined via back-and-forth considerations: Shelah's recently-introduced $\mathbb{L}_\kappa^1$s, for $\kappa$ an appropriate uncountable cardinal. I don't know much about this, but it seems interesting and reinforces the importance of back-and-forth systems; see these slides of Villaveces.
Finally, note that the above are all examples of logics that are substantially stronger than $\mathsf{FOL}$; what about weaker logics? An interesting example is provided by equational logic (see these notes of McNulty). Birkhoff's HSP theorem gives a characterization of equational equivalence: we have $Th_\mathsf{EqL}(\mathcal{A})\subseteq Th_\mathsf{EqL}(\mathcal{B})$ (= every equation true in $\mathcal{A}$ is also true in $\mathcal{B}$) iff $\mathcal{B}$ is a quotient of a subalgebra of a power of $\mathcal{A}$. Put another way, a class of algberas is equationally axiomatizable iff it is closed under quotients, subalgebras, and products. Interestingly, an analogous result also exists for $\mathsf{FOL}$: the Keisler-Shelah theorem states that a class of structures is first-order axiomatizable iff it is closed under ultraproducts and ultralimits. However, while the original HSP theorem is easy to prove, Keisler-Shelah is quite difficult. One natural question here is whether analogous theorems exist for infinitary or higher-order logics. However, it's unclear what "analogous theorem" means exactly in this context.
