The word "interpret" is usually used to compare two theories (or structures) in the same logic, so I'm going to avoid using it entirely here. Also, for simplicity I'll adopt a Platonist perspective here and assume that we're living in some "true" mathematical universe with a "true" second-order-logic satisfaction relation $\models_2$.
Every $\omega$-model $\mathcal{M}$ of $\mathsf{ZFC}$ (for simplicity, let's ignore non-$\omega$-models, although they don't really change things very much) has a version "$\models_2^\mathcal{M}$" of the satisfaction predicate for second-order logic. There are two ways that this predicate differs from $\models_2$:
It's only applicable to structures inside $\mathcal{M}$. It makes no sense to ask whether $\mathcal{A}\models_2^\mathcal{M}\varphi$ for $\mathcal{A}\not\in\mathcal{M}$.
More significantly, even when $\mathcal{A}\in\mathcal{M}$, the predicate "$\mathcal{A}\models^\mathcal{M}_2$" is determined via reference to what $\mathcal{M}$ thinks is the powerset of $\mathcal{A}$.
This second bulletpoint is absolutely lethal. For example, there is a second-order sentence $\theta$ such that $\theta$ is ("truly") true in exactly the uncountable structures. By downward Lowenheim-Skolem, there is a countable $\omega$-\model $\mathcal{M}\models\mathsf{ZFC}$. We then have for example $$\mathbb{R}^\mathcal{M}\models^\mathcal{M}_2\neg\theta\quad\mbox{but}\quad\mathbb{R}^\mathcal{M}\models_2\theta.$$ In general, the version of second-order logic developed inside an $\omega$-model of $\mathsf{ZFC}$ may "get lots of stuff wrong." This is in contrast to first-order logic, which is appropriately absolute. Note that this is really about first-order logic, not $\mathsf{ZFC}$ specifically - passing to a stronger theory will not help at all.
There are many more observations. For example, the set $\mathit{Taut}_2^\mathsf{ZFC}$ of "$\mathsf{ZFC}$-provable second-order tautologies" $$\mathit{Taut}_2^\mathsf{ZFC}=\{\varphi\in\mathsf{SOL}:\mathsf{ZFC}\vdash\forall \mathcal{A}, \mathcal{A}\models_2\varphi\}$$ is computably enumerable. But the set of actual second-order tautologies is not c.e. ... and $\mathsf{ZFC}$ can prove that!
Of course it is consistent (assuming mild hypotheses) that there is an $\omega$-model $\mathcal{X}$ of $\mathsf{ZFC}$ which "computes second-order satisfaction correctly," in the sense that $\models_2^\mathcal{X}$ coincides with $\models_2$ on all structures in $\mathcal{X}$: if $V_\kappa\models\mathsf{ZFC}$ then obviously $V_\kappa$ has this property! But this doesn't make up for the above issues. Also, we might want more - e.g. that $\mathcal{X}$ computes $\mathit{Taut}_2$ correctly - and then things get massively more difficult: for example, if there is a measurable cardinal then we have $$\mathit{Taut}_2^{V_\kappa}=\mathit{Taut}_2\implies\kappa\mbox{ is greater than the least measurable cardinal},$$ and so on.
I've avoided talking about non-$\omega$-models since they really just make things weirder, and we already see the core issue emerge with $\omega$-models. For example, if $\mathcal{M}$ is a non-$\omega$-model of $\mathsf{ZFC}$, then $\models_2^\mathcal{M}$ differs from $\models_2$ in a third way: we will have things $\mathcal{M}$ thinks are second-order sentences which are not actually finite. (This even trickles down to the first-order side of things, where the absoluteness of first-order logic referred to above breaks down when we look at nonstandard sentences - see Hamkins/Yang, Satisfaction is not absolute, which explains the importance of the adverb "appropriately" a couple paragraphs above.)
