Edit: Maybe I wasn't clear my question is: If the model theory is expressible in a formal language is it a formal theory? If not why? If the theory of the models is not expressible in a formal language is not problematic for the theory models itself?
Maybe it is a strange question. The Theory of models is a formal theory of some languages $L$ of the first order? If Yes which one? If yes there is a $L$-structure that satisfies this language or in other words, is the theory of models consistent/satisfiable? Does it even make sense to ask these questions? It is a self-reference? If it is self-reference what guarantees us that the theory of modules is something "true"? If the Theory of models is not a formal theory of some languages $L$ it is not problematic to define concepts as theory, models, consistency, satisfiability, etc. with an idea that is not a formal "somethings"?