Games are classified by what you do in them.
The main thing you do is fight. There are platforms but jumping isn't the main thing, so it wouldn't be right to call it a platformer.
You don't shoot so it isn't an FPS. You don't role play so it isn't an RPG.
Dislike the term "Brawler" since that is more of a throwback term for stuff like Double Dragon and Final Fight.
You mean beat-em-ups? FPS means FIRST PERSON shooter. It has to be FIRST PERSON for it to be an FPS. Gears of War for example is NOT an FPS.
Smash Bros. isn't just about straight combat. It's also about items and has a lot to do with maneuvering. Blocking is rarely used. This is how I see those genres:
Fighting Games: Street Fighter, Tekken, Virtua Fighter, Mortal Kombat, Guilty Gear, etc
Shoot-Em-Up (sub-genre of shooter): Gradius, Ikaruga, Aero Fighters, etc
THIRD PERSON shooter (TPS) (sub-genre of shooter): Gears of War, Uncharted, Hitman, Freedom Fighters, etc
FIRST PERSON shooter (FPS) (sub-genre of shooter): Half-Life, Call of Duty, Battlefield, etc
Beat-em-up: Double Dragon, Final Fight, TMNT, Simpsons, X-Men, etc
Brawler (sub-genre of fighting games): Onimusha Blade Warriors, Smash Bros. TMNT Smash Up, Viewtiful Joe Red Hot Rumble, etc
These classifications make sense. There's further sub-genres and hybrids (like cover-shooter, which implies a TPS with a cover system)
Am I the only person that understands logical sets and subsets or something?
The reason I'd advocate calling Smash Bros. and games like it something other than "fighting games" is because it's not what people mean when they say "fighting games", it's a sub-genre, absolutely, but when something thinks of a fighting game they're thinking Street Fighter 2, not Smash Bros. They play ENTIRELY different. It's not to devalue Smash Bros or anything (I love smash bros.) it's to differentiate so people can understand what the game is like ... that's the entire point of genres ... If you start making a genre that means anything ... then the purpose of genres disappears.
Technically ... mathematic-logic speaking ... you could say Smash Bros. is a fighting game, because it's genre is a sub-set. But, for people ... it just makes it confusing (and that really only has use if you're doing something like abstraction which you can use to have sort of global association. i.e. if you like fighting games you might like smash bros. because it's like these base fighting games but is still different enough that it can be in a different genre)