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, and where we don't care about the distinction between different infinite cardinalities (i.e., number is something in $\omega$ or $+\infty$)?

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 of [Bárány, Kaiser, and Rabinovich - Expressing cardinality quantifiers in monadic second-order logic over chains][1], so the only interesting case is where $\phi$ and $\psi$ both have finite numbers of satisfiers. (The same paper's Corollary 18 shows that we can define the uncountability of the number of sets satisfying a formula in MSO on $(\mathbb N,<)$ so given the Continuum Hypothesis, we don't need the restriction that "number" doesn't care about which infinite cardinality we have, and the finite case is still the only case to consider.)

  [1]: http://www.math.tau.ac.il/~rabinoa/pub/cardinality-quantifiers-jsl.pdf