Are there keywords to search for to read more? This is the first time I've heard about this! This is super interesting, what are a few well known other stable matches people deeper into this talk about?

The set of stable matches for a given problem seems to be called the lattice of stable matching.

https://en.wikipedia.org/wiki/Lattice_of_stable_matchings

The problem is it's a lattice and not fully ordered. There are cases where there are two or more pairings that are worse for women and better for men, but we cannot determine which pairing is superior to the other overall.