Recall that a structure $\mathcal{M} = \langle M, I^\sigma_M \rangle$ in a signature $\sigma$ is categorically axiomatized by a second-order theory $T$ when, for any $\sigma$-structure $\mathcal{N} = \langle N, I^\sigma_N \rangle$, $\langle N, \mathcal{P}(N), I^\sigma_N \rangle \vDash T$ just in case $\mathcal{N}$ is isomorphic to $\mathcal{M}$.

It is fairly easy to find a structure in a finite signature that is categorically second-order axiomatizable but not finitely categorically second-order axiomatizable. Add a single function symbol $f$ to the language of second-order arithmetic, and choose a non-second-order-definable $\zeta: \mathbb{N} \rightarrow \mathbb{N}$. Then consider the theory $T$ that adds to the axioms of second-order arithmetic ($\mathsf{Z}^2$) the sentence $f(\bar{n}) = \overline{\zeta(n)}$ for each natural number $n$, where $\bar{m}$ is the canonical numeral for $m$. (I owe the idea for this example to Andrew Bacon.)

This theory $T$, however, is not recursively axiomatizable. Is there a structure in a finite signature that has a recursive categorical second-order axiomatization but no finite categorical second-order axiomatization?

I believe that it is possible to find a recursively axiomatizable second-order theory $T$ whose spectrum (i.e., the set $\{\kappa \in \mathsf{Card}: \exists \mathcal{M} (\mathcal{M} \vDash T$ and $\vert \mathscr{M} \vert = \kappa)\}$) is shared by no finitely axiomatizable second-order theory, using partial truth predicates. (Consider the theory with $\mathsf{Z}^2$ relativized to some predicate $N$ and $\{$"The cardinality of the non-$N$s is not $\Sigma^1_n$-characterizable"$: n \in \omega\}$.) But I cannot see how to turn this into a categorical theory.