it's possible to extend the analogy to the factorization of polynomials over finite fields $\mathbb{F}_q$ (see this blog post for details; one needs to take $q \to \infty$ for the statistics to match, and in the post I only verify convergence in the sense of moments and am a bit sloppy). In this setting the permutation is Frobenius. But I don't think there's an analogous statement on the number field side.
I think results like this should be thought of as central limit-type theorems more than anything else; there are certain kinds of statistics that occur universally in certain general situations which otherwise don't necessarily have much in common.