Even so a model of ZFC might be given together with a collection of classes, only the sets should count as real. So if one would modify the collection of classes without modifying the sets, it would still be the same model. The formula $\forall y (y\in X)$ defines a class $X$ which cannot be modified at will, but a formula using impredicative quantification over classes will not define such a fixed class.
I feel this is similar to how a manifold in differential geometry can be embedded into a simpler but larger space. The simplest way to describe a manifold like the $n$-sphere might be its obvious embedding into $\mathbb R^{n+1}$, but the embedding is still arbitrary and unimportant. The simplest way to describe a manifold (like a projective space) might also be as equivalence classes of a simpler but larger space. This sort of external description is just as arbitrary and unimportant as the external description provided by an embedding. The models of ZFC provided by the model existence theorem are given as equivalence classes of simple syntactic terms, so this analogy still works.