Question

Why does the fog resembles prufrock's mental state?

Answer 1

The main purpose would be to simply be poetic, and use an interesting metaphor for the fog. Cats are known to be quiet, stealthy, adept, and able to scale and inhabit places that other creatures can't. So, comparing the fog to a cat is an apt description because fog too is silent, stealthy, and creeps into places that others can't.