Are there structures in a finite signature that are recursively categorically axiomatizable in SOL but not finitely categorically axiomatizable? 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.
 A: Sure - in fact, we can find a countable (and pretty concrete) example.
Let $\mathcal{N}=(\mathbb{N};+,\times)$ be the standard model of (first-order) arithmetic and let $R\subseteq\mathbb{N}^2$ be the binary relation given by $(a,b)\in R$ iff the $a$th $\Sigma^1_b$ sentence (in arithmetic, according to some standard enumeration) is true in $\mathcal{N}$. That is, $R$ codes the second-order theory of $\mathcal{N}$.
The structure $(\mathbb{N};+,\times, R)$ is second-order recursively axiomatizable: we use a single second-order sentence to fix the $\{+,\times\}$-reduct, and then recursively say that each column of $R$ codes the appropriate set. By the non-collapsing of the second-order hierarchy, this can't be done by a single second-order sentence.
Note that this trick can be generalized to a wide variety of logics. For logics like $\mathcal{L}_{\kappa,\lambda}$ which don't "fit" into $\mathbb{N}$ in a reasonable way, we need to $(i)$ shift to a different "base" and $(ii)$ possibly adopt a different notion of "recursive," but an analogous result should hold.
A: There is even an example of a cardinal  $\kappa$ and an r.e. categorical second-order theory $T$ such that for no finitely axiomatized second-order theory $U$, the spectrum of $U$ has $\kappa$ as its least element.
Let $T_0$ be the theory $\mathsf{ZC}_2+\forall x\exists \alpha(x\in V_\alpha)$, here $\mathsf{ZC}_2$ is the second-order version of Zermelo set theory where in the separation scheme we allow formulas without class quantifiers, but with class variables. It is easy to see that the second-order models of this theory are precisely $(V_\alpha,\in)$, for limit $\alpha>\omega$. Further I denote by $\mathcal{L}$ the set of sentences of the language of this theory.
Let $T$ be the extension of $T_0$ by

*

*$\forall \alpha (\exists \varphi\in \mathcal{L}) (((V_\alpha,\in)\models_2\varphi)\land \forall \beta<\alpha ((V_\beta,\in)\not\models_2\varphi))$.

*$\varphi\to\exists \alpha ((V_\alpha,\in)\models_2 \varphi)$, where $\varphi$ ranges over $\mathcal{L}$.

Obviously the only second-order model of $T$ is $(V_\alpha,\in)$, where $\alpha$ is the least ordinal such that for any $\varphi\in\mathcal{L}$ if $(V_\alpha,\in)\models \varphi$, then for  $(V_\beta,\in)\models \varphi$, for some $\beta<\alpha$. Clearly, $\alpha>\omega^2$ and hence the cardinality of this model is $\beth_\alpha$, which will be our $\kappa$.
Suppose for a contradiction that there is a second-order sentence $\varphi$ such that the smallest model of $\varphi$ is in the cardinality $\beth_\alpha$. Let $\varphi'$ be the naturally constructed sentence of second-order set theory expressing that there exists a class-model of $\varphi$. By construction $(V_\alpha,\in)\models_2\varphi'$, but for no $\beta<\alpha$ we have $(V_\beta,\in)\models_2\varphi'$, contradiction.
