Its a fighter. As Ghost mentioned it uses multiple ways to determine victory, but the net result is a fighting game.
As far as the platforming, there needs to be some clarification/understanding of meanings.
Platforms/Platforming are a set of elements and mechanics that pertain to conveyance throughout the game world. Some games dumb down the platform elements as a way to simply move from one objective to another or one battle scene to the other. Ratchet and Clank is a good example of this.
Now for a game to be considered a platformer, it has to be a game solely about getting from point A to point B. The puzzles, obstacles, level design, physics, and character abilities/controls are all designed and fitted to create a game about going form A to B. Using the example of a car... games can have cars, but not all games are racing games. Games can have platforms, but not all games are platformers. Mario, Mega Man, 'splosion man, N+ and LBP are examples of platformers.
With that in mind Brawl cannot be considered a platformer. The Emissary may have some minute dumbed down platforming elements, but the character controls are not re-tuned for that kind of game and the results are the floaty and honestly terrible moments between battles. They same can be said for the physics which dont offer themselves for a platformer experience. The game has no puzzles or obstacles.
Considering Brawl to be a platformer would be like calling GTA a racing game.
Sorry for the length, i can be passionate about platformers
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