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: L_omega_omega (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 t... (read more)
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.
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: L_omega_omega (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 t... (read more)