Heroes of Annihilated Empires has the option. You can choose to go into a match either as a builder or a solo champion. Goal is still annihilation, but the mechanics of it can be somewhat distinct (builders eventually get a hero as well, but later in the game and substantially weaker than the solo champ). The individual factions are fairly different, as well, though the nature of the game itself (everything dies in droves, constantly) makes that somewhat difficult to really notice.
Few of Arcen Games' other works have the same sort of asymmetry to their player/npc interactions, as well. Last Federation, Bionic Dues, even the Valley stuff... it's pretty common to their designs.
Pretty sure Distant Worlds can be set up similar to that third option, as well -- there can be nation specific victory conditions, iirc. Pretty sure there's a good handful of other 4x games that work like that... it's not exactly common, but it's not particularly unknown, either.
Several of the Civ games offer similar options, though anyone can pursue any particular one, rather than a particular faction being locked in to certain victory conditions. Still, you had disparate victory conditions available.
Giants: Citizen Kabuto probably counts, too. Fairly radically different factions, albeit with fairly similar goals.