I'm going to quickly summarize what I commented under Matt Colville's video I think a potential framework is preference of narrative types, and preference for game mechanics. I mentioned a few archetypes that I've seen that seem to be pretty common.

I talk about the Tactician GM, who's campaigns tend to be tests of skill and understanding of a given ruleset, think more dungeon crawls and mega dungeons. They tap into the wargame simulation roots of the hobby. The mechancial highlight system of this would be D&D or Pathfinder.

I also mention the Philosopher GM, who tends to focus on character's beliefs and convictions and how they operate in a messy world, what they're willing to sacrifice when push comes to shove. Think gamified trolley problems and philosophy 101. The mechanical highlight systems for this would be something akin Mage the Ascension/Awakening or Ars Magica.

And finally in my writeup, theres the Socialite GM, who specializes in the interpersonal dynamics and conflicts, think court politics, social maneuvering, or interpersonal melodrama. The mechanical highlight for this would be Vampire the Masquerade, especially any campaign focused heavily on Camarilla internal politics.

I think there may be a sort of parallel between this discussion of what is a GM's style and the Adam Neely video about what makes musical genres distinct.