A finitely axiomatizable consistent second-order theory without a model The completeness theorem fails for second-order logic. This question has some nice examples of consistent second-order theories without models. But non of them is finitely axiomatizable, at least those examples use infinitely many axioms.

Are there consistent finitely axiomatizable second-order theories without models, or is it possible to prove a completeness theorem for these theories?

 A: There is an ambiguity here which arises because of your focus on finite counterexamples - what proof system are you using?
A proof system for a logic is generally taken to be at the very least a relation between sets of sentences and individual sentences which is monotonic (more axioms prove more theorems) and finitely based (if $T$ proves $\varphi$, then some finite subset of $T$ proves $\varphi$). This last condition seems out of place in the context of second-order logic, where compactness fails wildly, but captures the idea that proofs are finite objects. This rules out the possibility of complete proof systems for incompact logics, leading to examples as in the linked question, but leaves open the possibility of proof systems which are complete when restricted to finite sets of sentences.
Now the barrier becomes comparability theoretic - we probably want the proof relation to be c.e., but this prevents completeness even for finite sets of sentences in second order logic. So regardless of what proof system you use, there will be consistent finite unsatisfiable sets of sentences - but to give a concrete example we would need to specify a proof system first.
