Is second-order logic *with standard semantics* necessary to categorically characterise the natural number structure?

One can prove that any two models of Dedekind-Peano arithmetic are isomorphic (categoricity theorem), and, by invoking the isomorphism theorem, that one model of Dedekind-Peano arithmetic satisfies A iff any other model of Dedekind-Peano arithmetic satisfies A, for any sentence A (which gives us elementary equivalence).

My question: can one do the same with second-order logic and Henkin semantics or does one require standard semantics for this categoricity result?

I suspect that because Henkin semantics is almost the same as many-sorted first-order semantics, one cannot categorically characterise the natural numbers using Henkin semantics.