There is an important distinction to make. The second-order induction axiom is just that - an axiom. It can be interpreted in several ways.
In normal second-order arithmetic, $Z_2$, we do have the usual second-order induction axiom
(\forall X)\big [(0 \in X \land (\forall n)[n \in X \to n+1\in X]) \to (\forall n)[n \in X]\big ]
But $Z_2$ is usually studied with first-order semantics, and in that context it is an effective theory of arithmetic subject to the incompleteness theorems. In particular, $Z_2$ includes every axiom of PA, and it does include the second-order induction axiom, and it is still incomplete.
Therefore, the well-known categoricity proof must not rely solely on the second-order induction axiom. It also relies on a change to an entirely different semantics, apart from the choice of axioms. It is only in the context of these special "full" semantics that PA with the second-order induction axiom becomes categorical.
Now, if we fix a sound deductive system for $Z_2$, the change of semantics has no effect whatsoever on the formulas that are provable. So, even though $Z_2$ with full second-order semantics is categorical, for any sound effective deductive system there are still true formulas of $Z_2$ that are neither provable nor disprovable in that system.
This answers a question in the comments, "How can a categorical theory be incomplete?" The answer is that categoricity is determined both by the choice of axioms and the choice of semantics, while completeness in this sense is determined by the axioms and the choice of a deductive system. (Here "complete" means that the set of provable theorems is a maximal consistent set.) There is no reason why, in a general setting, categoricity should imply completeness. In fact, it doesn't.
No matter what semantics we wish to use, it is simply impossible - by the incompleteness theorems - to come up with any effective deductive system that extends PA and is complete in the sense of the previous paragraph. Moving to higher-order systems helps us prove additional true propositions about the natural numbers, but we can never find a deductive system to prove all of them.