The point is that you already assume there is some set theory in play when you talk about second-order logic.
In other words, you use first-order set theory to talk about second-order "everyday theories" like the natural numbers, etc.
Different models of $\sf ZFC$ can have wildly different sets which they regard as "the true $\Bbb N$". If $M$ is a model of $\sf ZFC$, then there are $M_0$ and $M_1$ which are elementary equivalent to $M$, but $M_0$ is countable, and therefore has only countably many "natural number objects", whereas $M_1$ is such that it has uncountably many "natural number objects". Certainly these are not isomorphic, even though the models themselves pretty much agree on "how things should look like".
So yeah, what we actually have is that first-order $\sf ZFC$ proves that the second-order theories of $\Bbb N$ etc. are categorical. Or, if you will, inside a fixed universe of set theory, $\Bbb N$ has a categorical second-order theory. But moving to a different universe might produce you with different theories and different models.