This isn't correct. While it's true that in second order logic the natural numbers admit categoricity, second order logic lacks axiomatic semantics. So yes, there is a single set which can be called the natural numbers in second order logic (namely the intersection of all sets that satisfy Peano's axioms), but this set has no interpretation.
You can adopt Henkin semantics to give the naturals an interpretation, which is still second order logic, but then you're back to lacking a categorical model of the naturals.
You can adopt Henkin semantics to give the naturals an interpretation, which is still second order logic, but then you're back to lacking a categorical model of the naturals.