Dreamcast - Soul Calibur
Xbox - Halo: Combat Evolved
360 - Gears of War
PS1 - Final Fantasy VII/Gran Turismo/Metal Gear Solid (impossible to pin down on one)
PS2 - This one is actually hard simply because of its phenomenally outstanding repertoire of games. I'd probably pick GTA3
PS3 - Killzone 2 (for better or worse, that trailer didn't help)
Gamecube - Resident Evil 4
Wii - Wii Sports
PC - Doom
If you want to boil it down by format:
Sony - no idea
Microsoft - Halo
Nintendo - Mario
Sega - Sonic
PC/Windows - Doom
It's odd that Sony doesn't really have a distinguishing series or character, but that's just down to its versatility, breadth and range of its games and first party titles. Someone mentioned Final Fantasy earlier, but that's a PC & 360 series too, and has been ever since 7 which came out over ten years ago.