The categoricity of the second-order Peano axioms simply means (in the context of ZFC set theory) that every model of ZFC contains exactly one thing (up to isomorphism) that it thinks is a model of those axioms.

However, different models of ZFC can have different $\mathbb N$s. Each model will think its $\mathbb N$ satisfies the second-order Peano axioms, but that is not generally true when viewed "from the outside". Oftentimes it will simply be the case that the collection that would be a counterexample to the induction axiom does not exist as a set in the model.

The categoricity of the second-order Peano axioms simply means (in the context of ZFC set theory) that every model of ZFC contains exactly one thing (up to internal isomorphism) that it thinks is a model of those axioms.

However, different models of ZFC can have different $\mathbb N$s. Each model will think its $\mathbb N$ satisfies the second-order Peano axioms, but that is not generally true (in standard semantics) when viewed "from the outside". Oftentimes it will simply be the case that the collection that would be a counterexample to the induction axiom does not exist as a set in the model.