As stated in the title, I'm studying ZKPs and I see they are just interactive proof systems that respect the zero-knowledge property. Now, if that's true, why aren't they used for IP problems, the complexity class of interactive proof systems that was originally introduced for them? I mean, isn't the proof mechanism interactive in ZKPs between Prover and Verifier? Then why are we talking about NP problems, which are the opposite of interactive?
The reason is that essentially, the class of languages in $\mathcal IP$ that are not in $\mathcal NP$ cannot be proven with an efficient prover. Since we are typically interested in the cryptographic context with efficient provers, the study of ZK typically focuses on $\mathcal NP$. (Note that when studying complexity and often in foundations of cryptography, we are certainly interested in inefficient provers.)
In order to see why outside of $\mathcal NP$ the prover cannot be efficient, note that any interactive proof with an efficient prover (given a witness) can be emulated by a machine who receives the witness and runs the proof locally, testing if the result is accept or reject. This is exactly the class of languages $\mathcal MA$. Under standard derandomization assumptions, $\mathcal MA = NP$. Thus, under these assumptions, any language that is not in $\mathcal NP$ can only be run with an inefficient prover (even giving it some witness). As such, it is of less interest when constructing protocols and the like.