A theory of the type you are asking cannot be that concrete, because:
By an old result of Victor Marek, it is consistent with the axioms of $\mathsf{ZFC}$ that the second order theory of every countable structure (in a finite vocabulary) is categorical. See this FOM-post of mine for a reference. It is well-known (and noted by Shelah) that this result outright fails in $\mathsf{ZFC}$ if the vocabulary is allowed to be infinite (see Lemma 1.2 of this paper by Lauri Keskinen).
In the above FOM-post, I conjectured that Marek's result can be extended to all Borel structures. This conjecture was verified by Harvey Friedman in this FOM-post.
In yet another FOM-post, Solovay showed that it is consistent with $\mathsf{ZFC}$ that as soon as a second order theory $T$ is both axiomatizable and complete, then $T$ is categorical. See also this other related FOM-post of Solovay.
For more on this topic, see this manuscript (Wayback Machine) by Lauri Keskinen, and this recent paper of Saarinen, Väänänen, and Woodin.