I always imagined a set as an "abstract container" which contains things said to be its elements. I treat the "abstract container" as part of the set and not as element of the set, and thus, one cannot say that my treatment is incompatible with extensionality axiom (i.e. a n-element set does not have the entity "abstract container" as an element, as its n+1-th element - it has it as a part).
This imaging suggest that there might exist a "structuralistic" set theory, where a set is represented as a structure - an abstract container together with its content, which I also call multitude. An abstract container without content is the empty set. One has to consider the abstract container alone as an urelement, since it is the couple (abstract container, content) which is called "set" and can have elements. "Symmetrically", one has to consider a proper class as a "content without a container", a "multitude". Are there multitudes which are not proper classes? I call multitude any "content without container", and I admit that there are also "finite multitudes".
It seems, it was Russell who first observed that the language of set theory cannot be used to model the plural of nouns of natural languages, because a set is always treated as one thing. Russell used two terms: "class-as-one" for sets and "class-as-many" for entities which could model the plural. I don't know what Russell proposed as a solution to this problem, but my solution lies in my intuitive "structuralist" set theory about which I shared above. Namely, I treat a class-as-one as a set and a class-as-many as a content without a container. Here is the intuition behind this: we can refer to a set by using its container and we say "one", we cannot refer to "a content without a container" and we say "many".
The axiomatic set theory of "Sets and Classes as Many" by John L.Bell sound to describe a structuralist set theory like the intuitive set theory I described above. Are there continuations of his reseach? Are there other axiomatic set theories which make the distinction between class-as-one and class-as-many?