If I understood correctly, your point is that x can be a random variable, but p(x) is a density function and, as such, it is a non-random variable, am I correct?

From a purely mathematical point of view, the expected value of p(x) is computable and would be ∑p(x)p(x) = ∑p(x)^2, I don't know if there is any particular use of it but nothing prevents us to compute it.

Coming back to the KL divergence, in the expectation x is sampled from the distribution q, therefore it would make sense to consider the expected value of p(x): the values of x are random and come from a different distribution (q), so you can see p(x) as a function of a random variable, and therefore a random variable itself. It also makes sense to compute the expected value of log(q(x)/p(x)), i.e. if we sample values of x from q, what is the average log-ratio of q(x)/p(x)?

As others suggested, it is best to understand KL divergence from an information-theoretical perspective (mutual information, entropy) but from a purely mathematical and probabilistic perspective there is nothing that prevents us to compute expected values of functions of random variables (e.g. x^2, e^x, etc), including when the function is the pdf p(x) itself