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Dan123 comments on Logical Pinpointing - Less Wrong
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Mainstream status:
The presentation of the natural numbers is meant to be standard, including the (well-known and proven) idea that it requires second-order logic to pin them down. There's some further controversy about second-order logic which will be discussed in a later post.
I've seen some (old) arguments about the meaning of axiomatizing which did not resolve in the answer, "Because otherwise you can't talk about numbers as opposed to something else," so AFAIK it's theoretically possible that I'm the first to spell out that idea in exactly that way, but it's an obvious-enough idea and there's been enough debate by philosophically inclined mathematicians that I would be genuinely surprised to find this was the case.
On the other hand, I've surely never seen a general account of meaningfulness which puts logical pinpointing alongside causal link-tracing to delineate two different kinds of correspondence within correspondence theories of truth. To whatever extent any of this is a standard position, it's not nearly widely-known enough or explicitly taught in those terms to general mathematicians outside model theory and mathematical logic, just like the standard position on "proof". Nor does any of it appear in the S. E. P. entry on meaning.
A few points:
i) you don't actually need to jump directly to second order logic in to get a categorical axiomatization of the natural numbers. There are several weaker ways to do the job: Lomegaomega (which allows infinitary conjunctions), adding a primitive finiteness operator, adding a primitive ancestral operator, allowing the omega rule (i.e. from the infinitely many premises P(0), P(1), ... P(n), ... infer AnP(n)). Second order logic is more powerful than these in that it gives a quasi categorical axiomatization of the universe of sets (i.e. of any two models of ZFC_2, they are either isomorphic or one is isomorphic to an initial segment of the other).
ii) although there is a minority view to the contrary, it's typically thought that going second order doesn't help with determinateness worries (i.e. roughly what you are talking about with regard to "pinning down" the natural numbers). The point here is that going second order only works if you interpret the second order quantifiers "fully", i.e. as ranging over the whole power set of the domain rather than some proper subset of it. But the problem is: how can we rule out non-full interpretations of the quantifiers? This seems like just the same sort of problem as ruling out non-standard models of arithmetic ("the same sort", not the same, because for the reasons mentioned in (i) it is actually more stringent of a condition.) The point is if you for some reason doubt that we have a categorical grasp of the natural numbers, you are certainly not going to grant that we can enforce a full interpretation of the second order quantifiers. And although it seems intuitively obvious that we have a categorical grasp of the natural numbers, careful consideration of the first incompleteness theorem shows that this is by no means clear.
iii) Given that categoricity results are only up to isomorphism, I don't see how they help you pin down talk of the natural numbers themselves (as opposed to any old omega_sequence). At best, they help you pin down the structure of the natural numbers, but taking this insight into account is easier said than done.
Generally, things being identical up to isomorphism is considered to make them the same thing in all senses that matter. If something has all the same properties as the natural numbers, in every respect and every particular, then that's no different from merely changing the names. This is a pretty basic mathematical concept, and that you aren't familiar with it makes me question the rest of this comment as well.