As with other branches of mathematics, for basic (naive) model theory, all we need is naive set theory, and the notion of a set is just as naive.
And as with other branches of mathematics, at some point, we do need some formal set theory, so we assume that the sets we are talking about live in the mathematical universe, governed by ZFC. In fact, many model theorists find it convenient to assume, beyond ZFC, that there is a proper class of strongly inaccessible cardinals, though this assumption is usually just a simplification (namely simplifying the notion of a monster model).
The only internal assumption about universes of models in model theory (besides those forced by being a set in the Von Neumann universe) is, as far as I can tell, that they are non-empty. One can do model theory with empty structures, but that forces one to make a number of unpleasant exceptions, while empty structures are clearly beside the point, so it is usually avoided.