Doom had multiplayer over lan that I would say fits in that definition. For me personally arena shooters are more about movement speed and how weapons/health are managed as power ups to be found and managed as a recourses. I don’t consider multiplayer to be a requirement for the genre basically.

I always preferred the term “Arena Shooter” but it may have more specific meaning than any game that vaguely has elements popular in fps games of the 90s that I think “Boomer Shooter” is usually trying to communicate.