In "Anatomy of Integers and Permutations", http://www.dms.umontreal.ca/~andrew/PDF/Anatomy.pdf, Granville gives a calibration of cycles of a permutation and prime factors of an integer. "We know roughly one out of evert $\log x$ integers up to $x$ is a prime, and that exactly one in every $N$ permutations on $N$ elements is a cycle, so we could try to calibrate by replacing $N$ when we measure the anatomy of a permutation with $\log x$ when we measure the anatomy of an integer."

Is there also a useful link between irreducible factors of a monic polynomial of degree $n$ over $F_q[T]$ and either cycles or prime factors?

A naive guess would be to say that the link between irreducible factors and cycles is trivial due to the similarity of the following theorems:

Prime Number Theorem for Permutations: A randomly selected permutation of $S_n$ will be an $n$-cycle with probability exactly $1/n$.

Prime Number Theorem for Monic Polynomials: The probability that a random monic polynomial $f\in F_q[T]$ of degree $n$ will be irreducible with probability $\approx 1/n$ when $q$ is fixed and $n$ is large (or if $n$ is fixed and characteristic of $F_q$ is large).