When we say "$(M,\in_M)$ is a (set) model of $\mathsf{ZFC}$" we are making a mathematical statement that $M$ is a set and $\in_M$ is a relation on it such that all of the $\mathsf{ZFC}$ axioms hold. That is an objective statement about what $M$ is in our external view of it, a bird's eye view, if you will.
Assuming standard set theoretical foundations, we can formalize this mathematical statement in the language of set theory and do our reasoning with our formal system there, e.g. $\mathsf{ZFC}.$ So now we have a formal sense of what '$M$ is a set' and '$\in_M$ is a relation' mean: $M$ is a set, i.e. an element of our domain of discourse, and the relation $\in_M$ is another set, specifically a subset of $M\times M.$ We must also formalize the notion of sentences and satisfaction in some way, but these are routine mathematical concepts, much simpler than basic analysis or topology, so can easily be expressed in set theory.
(Of course, by Godel's theorem, we will never be able to prove such a model exists from $\mathsf{ZFC}$, but we can still talk about it hypothetically, or even prove it exists if we are willing to make assumptions beyond $\mathsf{ZFC}$. Also, the formalization becomes difficult if we want to talk about proper class models rather than set models. Due to difficulties with defining satisfaction recursively, the whole thing breaks down. This is in accordance with Tarski's theorem, which implies it can't work in $V,$ so it can't work in general.)
People in the comments have cautioned against talking too cavalierly about what models 'think' about each other (in reference to an earlier version of your question). With our bird's eye view we can talk about all kinds interesting facts about $M$ (yes, including if it 'knows about' some other model, etc). If we're being realists here, we might think of the bird's eye view as the 'thoughts' of a (proper class) model $V$ that is the 'true' set theoretical universe, although of course we have some serious philosophical difficulties in deciding what this model thinks about, say, the continuum hypothesis. Or we could take a more formalist tack and think of the bird's eye view as formal mathematical developments about hypothetical universe of sets that we are making and proving in our foundational system.
We know that, when viewed externally, models of set theory can exhibit all kinds of curious behavior
- The model $M$ thinks the sets in the set $M$ are the only sets in the universe, whereas we know there are more. (This also subsumes your example: $M$ thinks the sets in $M$ form a definable proper class, whereas we know $M$ is actually a set.)
- The $\in_M$ relation that $M$ uses may not actually be the true membership relation, i.e. we may have, for some $x,y\in M,$ $x\in_M y$ holds while actually $x\notin y,$ or vice-versa.
- The set $\omega_M$ that $M$ thinks is the integers may include 'infinite' integers that are not in (what $M$ thinks is) the closure of the successor relation from $0.$ In other words, $M$'s integers may be nonstandard.
- $M$ may actually be a countable set. This leads to the oddity called Skolem's paradox. $M,$ being a model of $\mathsf{ZFC},$ of course proves $\mathscr{P}^M(\omega_M)$ is uncountable. But, since $M$ only knows about countably many sets, $M$ can only think countably many sets are members of $\mathscr{P}^M(\omega_M).$ The mystery disappears when we realize that $M$ may simply not include the bijection between $\omega_M$ and $\mathscr{P}^M(\omega_M),$ since that is how we define what countable is. More generally, $M$ does not have an 'accurate' impression of the cardinalities of various sets.
- (Not really curious behavior, but something to think about.) By the completeness theorem, for any statement that is undecidable in $\mathsf{ZFC}$ there are models where it is true and models where it is false. So, for instance, assuming $\mathsf{ZFC}$ is consistent, we know there are models where the continuum hypothesis holds and models where it doesn't, and that there are models of $\mathsf{ZF}$ where choice holds and models where it doesn't. So remember that, while we can assume with a straight fact that the things $\mathsf{ZFC}$ proves are true, we know it certainly doesn't prove all true things (many statements are provably undecidable).
All this odd behavior can be proven to actually occur, at least under the assumption there are any models of $\mathsf{ZFC}$ at all (or equivalently that $\mathsf{ZFC}$ is consistent).
Is this a problem? Well, if it is, it is pretty unavoidable. As a commenter remarked, this issue with cardinality is really more about first order systems than about set theory. The Lowenheim-Skolem theorem says that if there are any infinite models of a theory, then there are models of all infinite cardinalities. First order logic is fundamentally unable to control the cardinality of its models (just try writing down a first order statement that says 'the universe is uncountable'... you can't and the LS theorem proves it).
Systems based on (full) second order logic are more well-behaved in this regard, but introduce new problems: they no longer have a complete deductive system so there's no effective way to prove what their logical consequences are. Furthermore, to make sense of them, we need a set concept in the metatheory. So we wind up in a situation where many statements (like the continuum hypothesis) have a definite answer in the sense that either they is true in all models or false in all models, but the answer depends on what the answer is in the metatheory (so this makes no more progress than first order logic in deciding them).
Overall it's probably best to take the perspective that while $\mathsf{ZFC}$ proves sensible, arguably true, things about mathematics, models of $\mathsf{ZFC}$ can be a little weird and counterintuitive. The bottom line is that a model is just a set and a relation on it that passes a few tests to show they behave like how we envision a set theoretical universe and its membership relation behave. We don't necessarily need to read more into it than that.
