The following is probably just an ELI5 version of Dacyn's answer:
Just because you use a word (such as "set"), it doesn't mean that it has an unambiguous meaning.
Imagine the same discussion about "numbers". Can you subtract 5 from 3? In the universe of natural numbers, the answer is no. In the universe of all integers, the answer is yes. Is there a number such that if you multiply it by itself, the result is 2? In the universe of rational numbers, the answer is no; in the universe of real numbers, the answer is yes.
Here you probably don't see any problem. Some statements can be true about real numbers and false about rational numbers, because those are two different things. A person who talks about "numbers" in general, needs to be more specific. As we see here, defining addition, subtraction, multiplication, and division is still not enough to allow us to figure out the answer to "∃a: a × a = 2".
It's similar with the sets. The ZF(C) axioms are simply not enough to pinpoint what you actually mean by a "set". They reduce the space of possible meanings, sufficiently to let us prove many interesting things, but there are still (infinitely) many possible meanings compatible with all of the axioms. For some of those meanings, CH is true, for other meanings, CH is false.
Is there a "set" greater than ℵ0 but smaller than 2ℵ0? It depends.
Is there a "number" that is greater than 2 but smaller than 3? It depends.
What makes certain axioms "true" beyond mere consistency?
Nothing. What do you mean by "true" here? Matching our physical universe? That in general is not what math does. The natural numbers may already include some that exceed the number of particles in our universe. The real numbers are inspired by measuring actual things, but do we really need an infinite number of decimal places? On top of all that, sets are merely mental constructs. A set of {2, 8, 33897798} does not imply anything about our world.
Is there a meaningful distinction between mathematical existence and consistency?
No.
Can we maintain mathematical realism while acknowledging the practical utility of the multiverse approach?
Maybe I am missing some important aspect, but the "multiverse" seems to me just like a intuitively helpful metaphor, but the actual problem is more like this: is the natural number "2" the same object as the integer "2", the real number "2.0", the Gaussian integer "2+0i", the complex number "2.0+0.0i", etc.?
One possible approach is to say: those are different domains of discourse... uhm, let's call them parallel universes to make it intuitive for the sci-fi fans. The object in a parallel universe is a different object, but also in some sense the captain Picard from the parallel universe is a natural counterpart to our captain Picard. They are generally the same unless specified otherwise for the plot relevant reasons, just like "2.0" from the real number universe is the natural counterpart to "2" from the integer universe, except that the former can be divided by three and the latter cannot. (Some things do not have a counterpart in the other universe, etc.) This feels like a natural approach for real vs complex numbers, and probably like an overkill for natural numbers vs integers.
The assumption of different universes kinda goes against the Occam's razor; we could simply move all these objects into the same universe (different planets perhaps) and make a story about a spaceship captain from Earth and a spaceship captain from Mars. Now we don't have the concept of a natural counterpart, and the analogies need to make explicitly: the horses on Earth correspond to the giant six-legged lizards on Mars. There is a set of natural numbers, the set of real numbers, and a function N -> R which maps the object "2" to the object "2.0". More importantly, there is no such thing as "addition"; there are actually two different things, "natural number addition" and "real number addition", and we call the latter the extension of the former, if for each pair of natural numbers, the counterpart of their sum is the same as the sum of their counterparts. The question whether "2" and "2.0" are intrinsically the same object can become kinda meaningless, if we always talk about numbers qua members of one or the other set. They could be the same object, or they could be different objects; the important thing is what they do, i.e. how they participate in various functions and relations.
(This kinda reminds me of Korzybski's "Aristotelian" vs "Non-Aristotelian" thinking, where the former is about what things are, while the latter is about how things are related to each other. Is "2" the same as "2.0"? A meaningless question, from the non-A perspective. The important thing is what they do; how are they related to other numbers. The important facts about "2" are that "1+1=2" and "2+2=4" etc. We can show that we can map N to R in a way that preserves all existing addition and multiplication, and whenever we do so, "2.0" is the image of "2". And that's all there is.)
With sets, I guess it is similar. If we have different definitions of what a "set" means, is the empty set according to definition X the same mathematical object as the empty set according to definition Y? The question is meaningless, from the non-A perspective; but to avoid all the complicated philosophy, it is easier to say that one lives in the universe X, and the other lives in the universe Y, so they are "kinda the same, but not the same". But to be precise, there is no such thing as an "empty set", only something that plays the role of an empty set in a certain system. Some systems could not even have such role, or they could have multiple distinct empty sets -- for example, we could imagine a system where each set has a type, and the "empty set of integers" is different from the "empty set of reals", because it has a different content type.
(Now I suspect I have opened a new can of worms, like how to reconcile Platonism with Korzybski's non-A thinking, and... that would be a long debate that I would prefer to avoid. My quick opinion is that perhaps we should aim for some kind of "Platonism of function" rather than "Platonism of essence", i.e. what the abstract objects do rather than what they are. The question is whether we should still call this approach "Platonism", perhaps some other name would be better.)
I have spent a long time looking in vain for any reason to think ZFC is consistent, other than that it holds in the One True Universe of Sets (OTUS, henceforth). So far I haven't found anything compelling, and I am quite doubtful at this point that any such justification exists.
Just believing in the OTUS seems to provide a perfectly satisfactory account of independence and nonstandard models, though: They are just epiphenomenal shadows of the OTUS, which we have deduced from our axioms about the OTUS. They may be interesting and useful (I rather like nonstandard analysis), but they don't have any foundational significance except as a counterexample showing the limitations of what formal systems can express. I take it that this is more or less what you have in mind when you say
It's disappointing that we apparently can't answer some natural questions about the OTUS, like the continuum hypothesis, but Gödel showed that our knowledge of the OTUS was always bound to be incomplete 🤷♂️.
Having said that, I still don't find the Platonist view entirely satisfactory. How do humans come to have knowledge that the OTUS exists and satisfies the ZFC axioms? Supposing that we do have such knowledge, what is it that distinguishes mathematical propositions whose truth we can directly perceive (which we call axioms) from other mathematical propositions (which we call conjectures, theorems, etc.)?
An objection more specific to set theory, as opposed to Platonism more generally, would be, given a supposed "universe" V of "all" sets, the proper classes of V are set-like objects, so why can't we extend the cumulative hierarchy another level higher to include them, and continue that process transfinitely? Or, if we can do that, then we can't claim to ever really be quantifying over all sets. But if that's so, then why should we believe that the power set axiom holds, i.e. that any of these partial universes of sets that we can quantify over is ever large enough to contain all subsets of N?
But every alternative to Platonism seems to entail skepticism about the consistency of ZFC (or even much weaker foundational theories), which is pragmatically inconvenient, and philosophically unsatisfactory, inasmuch as the ZFC axioms do seem intuitively pretty compelling. So I'm just left with an uneasy agnosticism about the nature of mathematical knowledge.
Getting back to the question of the multiverse view, my take on it is that it all seems to presuppose the consistency of ZFC, and realism about the OTUS is the only good reason to make that presupposition. In his writings on the multiverse (e.g. here), Joel Hamkins seems to be expressing skepticism that there is even a unique (up to isomorphism) standard model of N that embeds into all the nonstandard ones. I would say that if he thinks that, he should first of all be skeptical that the Peano axioms are consistent, to say nothing of ZFC, because the induction principle rests on the assumption that "well-founded" means what we think it means and is a property possessed by N. I have never seen an answer to this objection from Hamkins or another multiverse advocate, but if anyone claims to have one I'd be interested to see it.