Well, ideally your spotify artist page should contain the genres you're tagged with, along with some info of how that came to be, and even a way to request a change to those. But they don't want people to game the system, and also not let competitors know how they do things (I'm guessing).
At least you get the genre info is in the web API, just not present on the artist page (you need a token for the query to work, but you can create that on dev page):
https://developer.spotify.com/console/get-artist/?id=10YNQq8...
Here are a few articles about it (there are several more): https://artists.spotify.com/blog/how-spotify-discovers-the-g... https://www.thefader.com/2020/12/02/escape-room-genre-spotif... https://www.thestar.com/entertainment/2016/01/14/meet-the-ma...