My question is not about the algebra of this thing, but related to formal set theory.
I think that usually one will read that the Brauer group of the field K is made of equivalence classes of algebras over K of some type. (Which exactly doesn't matter for my question.) ... For example in the wikipedia article Brauer group the elements of this group are (quote:) Morita equivalence classes of central simple algebras over K. Again for my question it doesn't matter what Morita equivalence is exactly - the wikipedia article goes into category theory (which I don't know well yet) and also mentions Galois cohomology (which I haven't studied - don't know even what it is) - but I know a IMHO more elementary definition of the equivalence relation concerned, a bit like the one in Structure of Algebras by A. A. Albert who talks about "similar algebras" which must have been a usual name at some time - he doesn't use the name Brauer group (instead just he just uses the expression class group of K), but I see from his description that similar alg. = Morita equivalent alg. because I also know the name Brauer group (must be from another source).
Now the set-theoretic problem is: when an equivalence relation is always true between isomorphic structures, then correpsonding equ. classes are PROPER CLASSES in the sense of say NBG set theory (as soon as one considers non-empty examples) and with such classes you are not allowed to form sets nor other (= proper) classes. I know of a version of ZFC set theory which contains a trick to go around this obstacle. The same could be done with NBG but I never saw it done - in both cases one uses extra symbols and extra axioms not contained in original ZFC / NBG ... and without this "tool" one has to introduce UNnatural definitions of ordinals & even worse of cardinals, but one could argue, that isn't really a problem.
But those solutions for specific equivalences (equipotent sets / isomorphic well-ordered sets) are not easy to extend by analogy (at least not in an manner obvious to me) to the case of the similarity (or Morita) equivalence of K-algebras. How is this done - I insist: not intuitive ideas, but formal set-theoretical definitions - ?