How do we know that the P versus NP problem is an NP problem itself?

Determining for any statement if there is a proof with $n$ symbols or less is an $NP$ problem (i.e. the proof can be checked in polynomial time with respect to the length of the proof and the statement), that's probably the sense in which they meant that "P versus NP is itself NP". However, it does not really make sense to assign a complexity class to proving any particular statement (such as $P\neq NP$), as that technically takes constant time.

The "P versus NP" problem is a single yes-no question: is $NP = P$? The correct answer is either "yes" or "no", we just don't know which. But the complexity of the answer is $1$.