Uh.. What Sonic game?

Genesis: Sonic & Knuckles.  (Specifically for it's lock-on technology, and patching sonic any of the 4(?) Versions of sonic 2, and sonic 3 when connected.)

PS2: Grand Theft Auto: San Andreas (Do whatever, and massive playground)

PS1: Mega Man Legends 2 (Great art style, gameplay, graphics, and voice acting... with the exception of mega man, who sounded horrible.)

PS3: Beyond: Two Souls. (Game shouldn't be running on this console.)

Xbox 360: Grand Theft Auto 5 (Game shouldn't be running on this console.)

Dreamcast: Marvel Vs. Capcom 2 (Enjoyed a long life, and kept the dreamcast at evo for 10 years, and at the time 56 characters was the most any fighting game had.)

Vita: Wipeout 2048 (Probably my most favorite Racing Game of all time.  But, I've never really sat down, and seriously considered a favorite racer.)

PS4: Fortnite  (Breaking down the wall, and being the first game to really make cross-play for all consoles standard.)

Saturn never had a new Sonic game. It had some Genesis ports of the original games and a lousy unfinished racing game. Sonic Xtreme was in development but canceled after many delays by a western team. Saturn's most defining game is its launch game. Panzer Dragoon. Some could argue NiGHTS but it took until the Wii to get a sequel. Panzer got 1 sequel on Saturn and a unique RPG spinoff with air battles. Panzer got one more sequel on OG Xbox and a remake this year.

I screamed out loud for joy when I heard this music play at Nintendo E3 2019. One of the best themes to any game ever.

May contain bias

NES: Super Mario Bros
SNES: Super Mario World
N64: Super Mario 64
GCN: Super Smash Bros. Melee
Wii: Wii Sports
Wii U: Splatoon
Switch: Super Mario Odyssey

GB: Tetris
GBC: Pokémon Crystal
GBA: Metroid Zero Mission
DS: Rhythm Heaven
3DS: Fire Emblem Awakening

PS1: Crash Bandicoot Warped
PS2: Persona 3
PS3: Metal Gear Solid 4
PS4: Bloodborne

PSP: Danganronpa (I guess)
PSV: Gravity Rush

Xbox: Halo 2
Xbox 360: Halo 3
Xbox One: idk

Master System: Wonder Boy in Monster World
Genesis: Sonic the Hedgehog
Saturn: NiGHTS Into Dreams
Dreamcast: Jet Set Radio

Turbografx 16: Castlevania Rondo of Blood
Neo Geo: Garou Mark of the Wolves

Although I'd put Halo:CE as the defining title I can see why Halo 2 may also be chosen by someone else as it was the second game that launched the multiplayer side of the franchise and effectively was the precursor to what the likes of COD:MW would bring to consoles, CE only had co-op play. Halo 2 effectively did what Goldeneye pulled off on the N64 and built on it immensely people were still logging into the game well into the 360's life for the multiplayer until the servers were turned off.

Going by the ‘one game that is synonymous with the console’;

Atari 2600: Space Invaders
Atari Jaguar: Alien vs. Predator

SEGA Mega Drive/Genesis: Sonic the Hedgehog
SEGA Dreamcast: SoulCalibur

Nintendo NES: Super Mario Bros.
Nintendo SNES: Super Mario World
Nintendo 64: Super Mario 64
Nintendo GameCube: Super Smash Bros. Melee
Nintendo Wii: Wii Sports
Nintendo WiiU: Nintendoland
Nintendo Switch: The Legend of Zelda: Breath of the Wild
Nintendo GameBoy: Tetris
Nintendo Virtual Boy: Warioland
Nintendo DS: New Super Mario Bros.
Nintendo 3DS: Super Mario 3D World

Sony PlayStation: Final Fantasy VII
Sony PlayStation 2: Grand Theft Auto 3
Sony PlayStation 3: Uncharted 2
Sony PlayStation 4: Grand Theft Auto 5

XBox: Halo Combat Evolved
XBox 360: Kinect Adventures

Missing, beside non big-5 systems, are the Atari 5200 and 7800 as well as Lynx, SEGA Master System, Saturn and Game Gear, Nintendo GameBoy Advance, Sony PSP and Vita and XBox One. I feel like none of these has a real ‘defining’ game, or at least I can’t think of one. Which speaks to their mediocrity of a lineup. Consequently most of those failed as big sellers. I read that Saturn has a defining game in at least Panzer Dragoon, so I guess I’ll accept that, but I don’t know. Some of the others, WiiU, PS4 and XBox360 have a defining game, though they are disappointing in this role for multiple reasons. A lot of the games are launch (-window) titles, I guess that makes sense.

Panzer Dragoon then.  As I said, never owned one so I really don't know what it had to offer.

On the Sega Saturn it's defining game is Nights even though the best game on it is Panzer Dragoon Saga but it's Nights that was the flagship of the platform it was an odd move pitting a new IP against what would become the defining game for the jump to 3D in SM64 but I think SEGA figured it's a new IP from the Sonic Team unfortunately for the game SM64 was a defining beast and buried Nights pretty badly in all aspects despite being a decently good game.

Can't be a defining game when it's only a cult hit. It took until the Wii to get a sequel and people finally realized NiGHTs just isn't very good. It never was. I have my Saturn copy but man it's just not a good game. Nice music and charm but it not great to play. It was a tech demo for Saturn. Panzer not only showed off the hardware but a great game as well. Then we got Zweii and Azure on the same platform. Not to mention a series that is still alive.

Nights is the game that most people think of when it comes to the Saturn that's why it's a cult hit and compared to it the PD series is niche as it sold much less so that's not a reliable argument to be making, a bad sequel doesn't stop something being defining either and is a null and void argument tbh because a game being defining for a platform is not determined by it's sequel. I still have my Saturn and all my games for it and prefer other games but Nights is the defining game for it like it or not as it's the flagship for the platform and it's analogue controller, yeah it hasn't aged well but then not many Saturn games have.

PD is still around because it's not hard following up an on rails shooter while Nights is a unique concept that was only ever going to serve as a one off game.

In my experience in SEGA communities plural the last 20 years on forums they always bring up PD when thinking Saturn. NiGHTS is typiclly the AND game.