The premise of your question seems to be that mathematicians are generally agnostic about the truth of the parallel postulate, whereas there's a live controversy about whether the continuum hypothesis is true or not. I think you've got that almost exactly backwards.
In geometry, you're right that everybody knows that there are structures that satisfy the Euclidean axioms and other structures that satisfy the axioms of hyperbolic geometry. And there's a general agreement that both kinds of structures exist equally well, in whichever sense the speaker thinks mathematical structures "exist" at all.
However everybody still agrees that Euclidean geometry is the one we're usually interested in. It's the one that, by tacit agreement, everyone will understand a statement about lines and circles to refer to unless it is uttered in a context that explicitly refers to hyperbolic geometry.
This is the closest thing to universal mainstream agreement that Euclidean geometry is the real one I can imagine, short of making a claim about the shape of the physical world we live in. And if we go the latter route and ask the physicists, the general theory of relativity tells us that neither axiomatic system describes the real world exactly (and plenty of actual astronomical observations back that up).
(By the way, note that we can describe and work with the Euclidean space and plane entirely without reference to the classical axiomatic description of it -- namely as $\mathbb R^2$ and $\mathbb R^3$ with appropriate algebraic definitions of what we mean by a line, circle, etc.)
What about set theory, then? Rather than an active controversy, it looks to me like a vast majority of contemporary set theorists are content with saying that CH is undecidable from the axioms and that's the end of the line as far as philosophy goes. I don't see anyone wasting much breath (or ink, or bytes) on seriously arguing that CH is necessarily true, or necessarily false in some philosophical sense.
Paul Cohen, who proved that CH is not provable in ZFC, did opine at the time that he felt the continuum hypothesis was "obviously false". But that was 50 years ago, and very few of his successors seem to share his assessment, or even be interested in having one of their own.
In general set theorists are pretty wary of suggesting that set theory even has a "intended interpretation" that will give an absolute truth value to the CH -- especially compared with the ease with which everyone agrees that the Euclidean space/plane is the intended interpretation of geometry. It is generally acknowledged that the intuition that set theory speaks about some Platonic universe of actually existing sets is useful and beneficial (not least because without it there seems to be scant reason to care), but it is also generally acknowledged that arguments that rest on it tell more about the speaker's thought patterns than they tell about any kind of objective truth.
We know -- at least assuming that ZF is consistent, and otherwise why bother at all? -- that both ZFC+CH and ZFC+!CH have models, but in either case the models we get in that way are definitely distinct from whichever intended interpretation we might believe in.
So "set theory scholars argue about whether CH or ¬CH is true" is about as far from what I see at it could possibly be.
(JDH's multiverse concept seems to me like it's not so much an attempt to propose a definite answer to the questions you envisage, as it's a plan to give "no, we really, really don't need to waste time arguing these questions" a backing that feels more intellectually weighty than simply pointing to our decades of failures to get something useful out of philosophical musings in this area).