Is there a way of saying in second order arithmetic that the number of sets $X$ such that $\phi$ equals the number of sets $X$ such that $\psi$, where $\phi$ and $\psi$ are formulas with $X$ free? I.e., can we define the equicardinality quantifier $\#_X \phi = \#_X \psi$ for sets with the resources of SOA?

We can define the concept of there being finitely many $X$ such that $\phi$ in second order logic (or more generally monadic second order logic with an infinity predicate for sets) using the trick in the proof of Proposition 7 here, so the only interesting case is where $\phi$ and $\psi$ both have finite numbers of satisfiers.