This question has been moved to philosophy.stackexchange.com
I'll try to abbreviate it here: the question asks about the "informal notion" that the fragment of $\text{ZFC}$ that is axiomatized by axioms of: Separation schema, Collection schema, Set Union, Power set, and Infinity; can be said to be true of? This theory permits all kinds of abhorrent violation of extensionality and foundation, so what kind of entities suits such a theory? For example David Lewis in his book (Parts of Classes) had given an interpretation for classes as mereological aggregates of atomic labels of mereological aggregates. Now though he himself had postulated the principle that identifies parts of class as subclasses of classes, yet still in principle we can think of weaker labels that violate extensionality, i.e. we can think of having two distinct labels for the same aggregate, also labels can explain all kinds of circular membership. So it seems that atomic mereology + a singleton labeling function, can provide some intuitive envisioning into that fragment.
Question: had there been comparatively similar ideas that can explain intuitively such flagrant violations of Extensionality and Foundation?