I don't think that Peano axioms are sufficient for defining $\mathbb{N}$. For one thing, Peano axioms form a theory, not a particular structure. Additionally, the theory of Peano arithmetics is not complete, so it even doesn't define a structure up to elementary equivalence.

On the other hand, there is a standard model of Peano arithmetics in the standard set theory (ZFC), namely the countable infinite ordinal – as described in other answers.

Also, you can easily construct a structure that is different but isomorphic to $\mathbb{N}$. But that is trivial and matematitians often think of structures “up to isomorphism”. So they often do not make difference for example between zero as a natural number and zero as an integer, even though they are strictly speaking different objects (different sets) according to the standard set-theoretic formal construction.

I don't think that Peano axioms are sufficient for defining $\mathbb{N}$. For one thing, Peano axioms form a theory, not a particular structure. Additionally, the theory of Peano arithmetics is not complete, so it even doesn't define a structure up to elementary equivalence.

On the other hand, there is a standard model of Peano arithmetics in the standard set theory (ZFC), namely the smallest infinite ordinal – as described in other answers.

Also, you can easily construct a structure that is different but isomorphic to $\mathbb{N}$. But that is trivial and matematitians often think of structures “up to isomorphism”. So they often do not make difference for example between zero as a natural number and zero as an integer, even though they are strictly speaking different objects (different sets) according to the standard set-theoretic formal construction.