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(Later edit - tried to clarify a couple of vague places concerning interpretations of theories that became evident in comments (thanks to Andrej Bauer, Mauro ALLEGRANZA and Emil Jeřábek). (To closers and downvoters: may I humbly direct your attention to the soft-question tag down there...))

This is an ideal question: I don't even know what I am asking

It has been inspired by reading What "metatheory" did early set theory/logic researchers use to prove semantic results?

Specifically by

The modern approach seems to be, usually, to interpret a "model" specifically as a set in some other (typically first-order) "set metatheory." So when we talk about a model of PA, for instance, we typically mean that we are formalizing it as a subtheory of something like first-order ZFC. Models of ZFC can be formalized in stronger set theories, such as those obtained by adding large cardinals, and etc.

Indeed. It makes me wonder: there are all kinds of theories; there is a very general notion of interpretation of one theory in another; nothing about sets comes into play so far. Then all of a sudden, when one says "model", the first thing comes to mind is "a set with blablabla...". Why?

I mean, is there a rigorous way to distinguish some theories among all others by some formal property which (a) guarantees that any sensible notion of model is subsumed by interpretability in a theory with this property and (b) every theory with this property is equivalent (in some rigorous sense) to "a theory of sets"? What does this latter even mean?

Are there any other such "all-encompassing" classes of theories? This last question might seem senseless since any two "all-encompassing" theories must be mutually interpretable and thus be equivalent to each other (I think, at least in some sense?). But still maybe there is some notion of "meta-encompassing" that makes some "all-encompassings" more "all-encompassing" than others...?