Urban fantasy tropes go way back to the pulps and comics of the early 20th century. Weird Tales was publishing hard-boiled PIs vs. supernatural horror very early, and Dr. Occult (among others) appeared in the comics in the 30’s. But pretty much all of it was marketed as horror.
When the horror boom collapsed in the 80’s, people started calling their horror “dark fantasy” to separate it from the serial killer and splatter punk stuff. At the same time, Charles de Lint was doing his thing, and War for the Oaks made its big splash.
I have no idea why it got the name “urban” though, except maybe from urbanites’ bias that city=modern and rural=the past.
But I don’t think the genre hit its stride until Buffy came along. I mean, there were people writing it before, but Buffy seemed to have a powerful effect on the readership. Readers who loved the shows began snapping up the books.
What’s more, BtVS showed how powerful and effective paranormal romance plots could be, and it enshrined the new story structure of the UF, which was a break from the de Lint style. That was:
Horror fiction + protagonist with agency = urban fantasy.
It seems to me that the genre has sort of backed into the crime thriller format (I’ve said “mysteries” before, but that was stupid of me. Mystery novels are usually extended series of interesting conversations. Thrillers have more violence, and it’s the thriller that UF has emulated.)
Once the protagonist gets juiced up the vampires and werewolves become more like thriller-ish super-criminals, and the protag no longer flees in terror. That just pushes the story toward this:
Awful event -> What caused this? -> investigation <--> violent clashes <--> plot twists | (until finally) extended climactic battle.
Which is a classic thriller style.
I’m not an expert in any of this, but this is how it seems to me.