Let $T=\{{T_i:i\in I\}}$ be a set of theories indexed by and index set $I$. Assume that each theory is satisfiable, and that for any model $m$, $m$ models $T_i$ for some $i\in I$ iff $m$ models $T_j$ for any $j\in I$. Equivalently, all the theories in $T$ have the same models.
Let $S=\cap_{i\in I}T_i$ be the intersection of all the theories of in $T$. Must all models that model $S$ also model $T_i$ for any $i\in I$?
No. Consider for example $T_1=\{\forall x\forall y(x=y)\}$ and $T_2=\{\forall x\forall y(y=x)\}$. Then the models of $T_1$/$T_2$ are exactly the one-element structures; but $T_1\cap T_2=\emptyset$ and every structure is a model of $\emptyset$.
We can fix this by looking at deductively closed theories, but then the answer is trivially yes: two deductively closed theories with the same classes of models are equal.