Allegience can involve any number of teams with any members of each team potentially in a game. Free-for-all is simply a number of teams equal to number of players.
abilities can be organized anywhere. You could teleport diagonally by pressing a button then push specific objects away with another. Use your imagination.
BRINK uses well-known fps conventions. Team Fortress 2's classes (which originated in the quake 1 team fortress mod), Call of Duty 4's shooting (Ironsights, sprinting, recoil, etc., which has also existed since splash damage's own Wolfenstein: enemy territory), and mirror's edge's free-running (literally all the moves from that game are usable by the light weight in BRINK). Grenades, scopes, silencers, etc. However, BRINK's weapon stats are actually truthful, abilities depend on the supply meter and cooldowns, and objectives have time limits all of which creates a great balance through time delays. I've loved the way enemy territory games have organized these elements since they started.
As for Solo? Bots-only. co-op? that exists in any team game just by being on a team, but in BRINK it's human team vs bot team. multiplayer? The heart of this game which is a complexly-designed multiplayer system.
How about lone wolf? Call of duty players never use teamwork it seems, but team fortress 2 forces it by forcing each class to play a certain way. BRINK has teamwork in class abilities your own choice in combat with weight. Combined with objectives-only instead of kill-focused gameplay, teamwork has a similar flow to TF2.
You could say BRINK is a combination of single and multiplayer, but it's honestly a multiplayer game with cinematics for story setup. (Those implications of this meaning something really good or really bad, remove those from your mind immediately. This is fact-stating, not ranting or praise. People have to stop being so overreactive.)