i don't think we KNOW. in fact, the possibility of there being "uncountable models" and "countable models" highlights the difficulty of "talking about real things". the very encoding of information in language, limits us to something that is somewhat less than "what there is", so that we can describe it without describing it "fully" (which would be pointless, since the universe has already "described itself" better than we could).
i think that, at best, we "hope" the models we come up with are appropriate, insofar as they don't contradict our experience. there is nothing about the physical world that suggests it is even infinite, much less uncountably so. but our experience with things that appear to "vary continually" suggests that describing the universe (or some bits of it) as a continuum is USEFUL. at some point, the formalism itself takes over, and research is carried on of questions that may never "apply" to the real world, but are "interesting". and why not? a painting does not have to have "photo-realism" to be meaningful to those who view it.
i suspect the relative view is more accurate: i doubt ANY formal system will ever "capture the known universe". i believe that we need the freedom to look at things under different (and perhaps even contradictory) lenses to get "the whole picture". these are, however, merely my own views, and many people disagree.