In his article "The Set-Theoretic Multiverse", Joel David Hamkins (as part of his reply to Donald Martin's argument that the set-theoretic universe is unique, found in "Multiple Universes of Sets and Indeterminate Truth Values") states that "...a second-order categoricity argument, even just for the natural numbers, requires one to operate in a context with a background concept of set." (p. 14 in the linked arXiv edition)
This got me to wondering, is this strictly required, or is a set-theoretic context for categoricity proofs just being assumed as part of the dialectic? I certainly grant that there is a notion of categoricity which seems intimately connected with your background set theory, since the proofs are carried out within that background set theory and since the standard semantics for second-order logic is a set-theoretic semantics.
But there are alternative semantics for second-order logic (e.g., the plural quantification interpretation of monadic second-order logic suggested by George Boolos) and while it might be the case that none of them are quite as good as the standard semantics--- for example, there is the trouble that relations give plural logicians ---it seems like this might open the door to presentations of second-order logic which don't rely on a background set theory.
Is the issue that the concept of "model" utilized in statements of categoricity is that of a set-theoretic model? If we had an alternate conception of model, one not tied to set theory, would a background concept of set still be presupposed by categoricity proofs? Are there any extant examples of putative categoricity proofs which do not operate within a set-theoretic background context?
In the interest of isolating one question, take the following to be the main question here: in what sense(s) do categoricity proofs require one to operate with a background concept of set and in what sense(s) is categoricity independent of set theory?