Is all of mathematics set theory in disguise?

No.

Say you've identified a mathematical theory $T$ with a model in $\mathsf{ZFC}$. It has multiple models in $\mathsf{ZFC}$, each with different this-object-is-that-set identifications. Not only are these identifications different by model, they also have at-the-set-level consequences that don't mean anything about $T$ itself. For example, some but not all models of $\Bbb N$ in $\mathsf{ZFC}$ satisfy $m\le n\implies m\subseteq n$, where the first statement is about naturals but the second is about sets with which they're identified when specifying a model. It would be silly to write $2\subseteq 3$, or $2\in 3$ (these are examples of what @NoahSchweber's answer calls junk theorems), because these identifications don't tell us what $T$'s objects "really are". The only reason models are interesting is that, as long as you can find one, $\mathsf{ZFC}$ (or whatever baseline we picked) implies $T$ is consistent.

You could just as easily use something else as a "base" for mathematics, e.g. category theory.

It depends what you mean by "is."

There is certainly a sense in which we can "embed" all of mathematics inside $\mathsf{ZFC}$. Consequently, if we imagine a person who for whatever reason is only ever willing to talk about actual formal proofs from $\mathsf{ZFC}$, they would still be able to prove not-explicitly-set-theoretic theorems by proving the appropriate set-theoretic translation.

However, $\mathsf{ZFC}$ - and more generally the framework of set theory - is not unique in this sense. Personally I find it the most natural framework, but this is absolutely a subjective position and there are good arguments against it; the one I find most compelling is the "non-structural" nature of $\mathsf{ZFC}$, mentioned in J.G.'s answer, namely that when we implement a piece of non-set-theoretic mathematics in $\mathsf{ZFC}$ we wind up with "junk theorems" in addition to the theorems which are translations of meaningful results in the original context.

So is mathematics set theory? Well, it can be if you want it to - but that says more about your choice of style than about mathematics.

I would say that it's the other way round: most mathematical topics can be disguised in set theory, but their true nature is only revealed when the disguise is cast off. For a simple example, the set theoretic view of the permutation groups $S_n$ as a set of functions each function being represented as a set of pairs doesn't help much in elementary group theory.