According to 1 there are axioms of second-order arithmetic which categorically characterize the natural numbers up to isomorphism. The axioms are:
$$\forall x\,\neg(x+1=0)$$ $$\forall x\,\forall y(x+1=y+1\to x=y)$$ $$\forall X\left(\left[X(0) \land \forall y\left(X(y) \to X(y+1)\right)\right] \to \forall y\,X(y)\right)$$
Only the last axiom uses second-order logic. It is the induction axiom, which in first-order arithmetic is replaced by an induction axiom schema.
These axioms of second-order arithmetic rule out "non-standard numbers" which are compatible with first-order arithmetic. Such non-standard numbers are numbers which have infinitely many predecessors and are larger than any natural number.
But how does the second order induction axiom rule out non-standard numbers, in contrast to the first-order induction schema? Since the second-order axioms talks about "all properties" $X$, I assume non-standard models must have at least one property $\Phi$ which contradicts the second-order induction axiom. But what is this property $\Phi$?