I regard ZF (or better ZFC) as a (partial) description of the behavior of actual sets. The theorem you quoted says, in that context, that there is no set containing everything. In the same context, I might sometimes talk about classes, but I would regard such talk as an abbreviation for statements that are only about sets, as explained, for example, in Jensen's book "Modelle der Mengenlehre." In other words, I don't think of classes as actual entities.

Concerning urelements, I would use a slightly modified version of ZF to describe a world of sets and urelements; see for example the theory ZFA in Jech's "Axiom of Choice" book. The theorem you quoted still holds in the presence of urelements, and it still has the interpretation that there is no set containing everything.

If some people (not me) wanted to work with sets and proper classes as genuinely existing entities, they would probably use a theory like Morse-Kelley to formalize their ideas. The theorem you quoted is still available; now it says that there is no class containing everything (including all classes).

There are, of course, set theories in which the theorem you quoted is not true; Quine's New Foundations and its variants are the most prominent of these. Here there is a set of everything. Unfortunately, I have no idea what sort of entities NF is "intended" to describe; perhaps Holmes's consistency proof will eventually lead me to such an idea.