Let $T$ be a theory of a language $\mathcal{L}$ with no function symbols. Let $\mathfrak{A}, \mathfrak{B} \models T$. For all finite sets $X \subseteq A$ and $Y \subseteq B$, there exists a function $F : X' \to Y'$ such that $$X \subseteq X' \subseteq A,$$ $$Y \subseteq Y' \subseteq B$$ and $F$ is an $\mathcal{L}$-isomorphism between the substructure of $\mathfrak{A}$ that is generated by $X'$ and the substructure of $\mathfrak{B}$ that is generated by $Y'$.
Is there anything that can be said about models $\mathfrak{A}$ and $\mathfrak{B}$? Is there any set of sentences $\Sigma$ that is absolute between the two models? For example it is easy to see that they satisfy the same existential sentences.
As you have noticed, the condition implies that $\mathfrak{A}$ and $\mathfrak{B}$ satisfy the same existential sentences. It follows that they also satisfy the same universal sentences. And, of course, they will satisfy the same boolean combinations of such sentences.
This appears to be as far as you can go, however.
Note that the structures $( \mathbb{N} , < )$ and $(\mathbb{Z} , < )$ satisfy the given condition. However they differ on the sentence $( \forall x ) ( \exists y ) ( y < x )$, which is a sentence of the next logical complexity. Therefore the condition is not enough to imply that $\mathfrak{A}$ and $\mathfrak{B}$ satisfy the same sentences with even a single pair of alternating quantifiers.