Recall (page 15) Detlefsen's first question:

Button and Walsh distinguish two issues for philosophers investigating categoricity ([Button & Walsh 2016], 283): ‘determinacy of reference of mathematical language and the determinacy of truth values of mathematical statements.’ As in their paper we focus on the question of determining reference for theories that are intended to describe a single isomorphism type of a structure. With the normal background of ZFC, one can define the vocabulary τ, the domain A, and the interpretation of the τ -relations on A. This description unambiguously refers to a particular structure. The isomorphism type of A is the class of all τ -structures isomorphic to A. The number theorist Barry Mazur wrote,

Importantly, one can also ‘take the ham’. For example, use linear algebra to study a matrix group and apply the result to any isomorphic group (Chapter 4.7).

There are several ways that a serious issue arises. Attempting to reify the notion of isomorphism type strikes the obstacle that the collection of structures isomorphic to A is not a set. This can be resolved by bounding the set theoretic rank of the structures or considering the isomorphism class as definable in ZFC; either of these is adequate from a model theoretic perspective. Secondly, one can demand that the description be formulated in the original vocabulary and then the problem becomes categoricity. In this context, the study of categoricity begins with the choice of logic. We argue, working from our pragmatic notion of virtue, that categoricity is interesting for a few second 68 order sentences describing particular structures.

But this interest arises from the importance of those axiomatizations of those structures and not from any intrinsic consequence of second order categoricity for arbitrary theories.