PS1 - Crash bandicoot 2
N64 - Mario 64
PS2 - Gran Turismo 3
GC - Resident Evil 4
PS3 - Uncharted 2
PS4 - The Last Of Us

Titanfall

N64: Super Mario 64
Xbox 360: Halo 3

GC - The Wind Waker
DS - New Super Mario Bros.
Wii - Super Mario Galaxy
3DS - Super Mario 3D Land
Switch - Breath of the Wild/Astral Chain

New Super Mario Bros. is not my favorite DS game but it's my first and the first that came to mind right now. It's also the first video game I ever played.

NES - Super Mario Bros
Gameboy - Pokémon Red
SNES - Super Mario World
Genesis - Sonic the Hedgehog
PS1 - Crash Bandicoot
N64 - Super Mario 64
Dreamcast - Sonic Adventure
GameCube - Super Mario Sunshine
Xbox - Halo
DS - Mario Kart DS
Wii - Wii Sports
Vita - Persona 4 Golden
WiiU - Mario Kart 8
Switch - Mario Kart 8

There's not really one special title that comes to mind when thinking of the rest. Probably as I didn't play the major hitters like GTA on PS2 etc. I don't associate them with one game.

NES: Super Mario Bros.
SNES: Super Mario All Stars
N64: Super Mario 64
PS1: the Demo 1 disc
PS3: GTA V
PSP: Prince of Persia: time sands
PS Vita: God of war HD collection
Wii: Super Mario Galaxy 2
PS4: God Of War
Xbox 360: Red Dead Redemption
Switch: Super Mario Odyssey
Steam Deck: Hogwarts Legacy

GameCube: Mario Kart Double Dash!!

GameBoy: Pokémon Blue

GameBoy Color: Warioland 2

Nintendo 3DS: Super Mario 3D Land

Bonus: PC: GTA Vice City

Trying to list games for the systems I have played significant amount of time on.

NES - Super Mario Bros 3 , could just as easy pick the first Super Mario, but SMB3 is the first game ever I remember liking

SNES - Super Metroid , this is kind of weird since I have not played that game or even like the series. And I think more of JRPGs is the genre that defines the SNES, but somehow Super Metroid is the first game that comes to mind when I think of the system.

N64 - Super Mario 64

GC - Super Smash Bros Melee

Wii - Wii Sports

Wii U - Super Mario Maker, the only game where I found the gamepad thingy nice to have.

Switch - Nothing comes to mind, my view of the switch is more general than specialized so no one game define it for me

PS1 - Crash Bandicoot

PS2 - Kingdom Hearts

PS3 - Uncharted 2 Among thieves , or just any game from Naughty Dog

PS5 - Nothing stands out as defining so far

Dreamcast - Shenmue

PC - Counter Strike, another game I have not played that much or even like. But online FPS games is just PC for me

Game Boy - Tetris

Game Boy Color - Pokemon Gold/Silver/Crystal

GBA - Fire Emblem, I bought a modded GBA just to be able to play this game in a modern way

DS - Warioware touched

3DS - Nothing comes to mind, did not really like the system or the games on it that much

These are some that come to mind first to me. Not all are exclusives, and not all are my top game on the platform (though many are). A few I haven't even played.
NES: Super Mario Bros.
GB: Pokemon Red/Blue
SNES: Super Mario World
N64: Super Mario 64
GBC: Pokemon Gold/Silver
GBA: Wario Land 4
GCN: Super Smash Bros. Melee
DS: Nintendogs
Wii: Super Mario Galaxy
3DS: Animal Crossing: New Leaf
Wii U: Super Smash Bros. for Wii U
Switch: Super Mario Odyssey
PS1: Final Fantasy VII
PS2: Star Wars Battlefront II
PSP: Persona 3 Portable
PS3: Uncharted 2 Among Thieves
Vita: Persona 4 Golden
PS4: Marvel's Spider-Man
PS5: Astro's Playroom

Atari - River Raid/Megamania

NES - Super Mario Bros 3
SNES - DK/Mario Kart/Super Mario World/Mario RPG/Zelda LttP/Chrono Trigger/FF6...
N64 - Mario 64/Zelda OoT/...
GC - Mario Sunshine/Metroid Prime/Paper Mario: The Thousand year door/Zelda WW...
Wii - Mario Galaxy/Zelda TP...
Wii U - Super Mario 3D World/ Zelda BotW...
Switch - Mario Odyssey/...
GBA - Super Street fighter 2/...
DS - FF4/Rhythm Heaven/...
3DS - Luigi's Mansion Dark Moon/Kid Icarus: Uprising/...

Master System - Wonder Boy/ Astro Warrior/Double Dragon/...
Genesis - Castle of Illusion/ Sonic 2/Altered Beast/Golden Axe/Streets of Rage/...
Saturn - X-Men:Children of the Atom/Panzer Dragoon 2/Guardian Heroes/Sega Rally/...
DC - Shenmue/Sonic Adventure/Street Fighter 3rd Strike/ Grandia 2/...

PS1 - FF7/Street Fighter Alpha 2/Gran Turismo 2/...
PS2 - Kingdom Hearts1 and 2/Gran Turismo 3 and 4/Dragon Quest 8/God of War 1 and 2/Gradius V/...
PS3 - Uncharted 2/Gran Turismo 5 and 6/ The Last of Us (multiplayer)/...
PS4 - God of War/ Uncharted 4/...
PS5 - Elden Ring/...
PSP - Capcom Classics Collection Reloaded/ Kingdom Hearts: Birth by Sleep/...
PSVita - Murasaki Baby/Persona 4 Golden/...

X360 - Halo 3/...

Well, let's see then.
PS1: Final Fantasy IX (Not the first game I played on it, but the one I always associate with the console. Also, the best game in general)
PS2: Shadow of the Colossus (Nothing else quite like it on any other console at the time)
PS3: Valkyria Chronicles (The first truly great game I played on the console, made buying it worth the price of admission)
PS4: Bloodborne (The best game of its generation, and the best reason to own the console)
PSP: Crisis Core (Didn't play too many games on the console, but of the ones I did play this is the one I most associate with it)
NES: Mega Man 4 (Not the best game on the console, but the one I have the fondest memories of playing back in the day)
SNES: Final Fantasy VI (The best game of its generation, some of the console's most memorable moments come from this game)
N64: Blast Corps (Rare at their best/weirdest)
Gamecube: Tales of Symphonia (My introduction to the series, and some of the most fun I've ever had playing co-op with my friends)
Wii: Super Smash Bros. Brawl (I don't even want to think just how many hours my friends and I spent playing Brawl back in the day)
WiiU: Nothing comes to mind in particular. Mario Kart 8?
Switch: Breath of the Wild probably (It was so deeply ingrained in the console's early identity that it's impossible to separate the two at this point)
Xbox 360: Lost Odyssey (The only game I've ever actually finished on the console)