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). 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.

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.