Well, I don't think it's safe to say natural numbers "are" sets, but surely they are isomorphic to some collection of sets (and this allows them to be modeled as sets within set theory).

The important part about the construction of the natural numbers from axiomatic set theory is that it can be done, not that it brings us closer to the Platonic idea of numbers. It can of course be done in many ways (OP's post lists just two). There's no reason to believe any specific representation within set theory is the true order of the universe, but it is extremely useful and we should be glad it works so well.

Thanks, that was an interesting rabbit hole. Although I can't help but feel there's a philosophical map/territory confusion here. Like, sure, numbers can't possibly just "be" sets because there are many different models of the naturals in (ZF) set theory, even in higher order logics. But I feel like a Platonist would just counter that of course this is the case - we are simply modelling the properties of the "true" naturals with these sets, in the same way that differential equations can model the behaviour of fluids without "being" water. Nobody writes down Navier-Stokes and expects to get wet!

Could we just say that natural numbers can be represented using sets, and leave it there?

Natural numbers can also be represented by even natural numbers (e.g. the easy way where 2n represents n), but that doesn't tempt people to make metaphysical statements about natural numbers somehow fundamentally "being" even. There is no reason why a representation of natural numbers by sets should be any more tempting.

There are multiple represenation of rational numbers . n * p / n * q (for any n). So this problem is not unique to reduction of numbers to sets. Generally people say unique reduced form.

It's not a map/territory confusion. It's a warning about map/territory confusion. The problem is that we have only maps, so we cannot fully understand the territory. This is a major concern in mathematics, whose reason for existence is to understand things precisely, not just rough and ready.

I'm not convinced there is a coherent "territory" when it comes to mathematics. Sometimes (remarkably!) mathematical concepts neatly line up with some part of the universe, and we gain some insight into how it works. But often that part of the universe lies entirely within somebody's mind - mathematics as an hobby or aesthetic pursuit, for instance, occupied with questions of pure logic.

Different ways of thinking about the same thing can lead to conflicting truths even without set theory getting in the way. The standard model of the real numbers has no infinitesimally small elements, for instance, but the hyperreal numbers do, despite satisfying all of the same first-order properties.