Question will be a bit naive, so please, be kind.
Consider a first order theory, $\Gamma$ . Let $\mathcal{M}$ be the category of models for $\Gamma$. Consider $\sim$ an equivalence relation on $\mathcal{M}$.
Is $\cal M/{\sim}$, in any sense, the category of models of any first order theory?
Let $\mathcal{M}$ be the category of Kan complexes. (The morphisms are the morphisms of simplicial sets.) Kan complexes are models for a many-sorted first-order theory. Let $\sim$ be the relation of simplicial homotopy. It is well known that $\mathcal{M} / {\sim}$ is equivalent to the homotopy category of CW-complexes, and a theorem of Freyd shows $\mathcal{M} / {\sim}$ does not admit a faithful functor to $\mathbf{Set}$. In particular, it cannot be the category of models for any first-order theory.