If I had to define a whole system by one game, then here are my picks. Note that some of these games I don't even like, even if they are on a system I like.
Atari 2600: Pitfall
Genesis: Sonic the Hedgehog
NES: Super Mario Bros.
SNES: Donkey Kong Country
N64: Mario 64
GCN: Smash Bros Melee
Wii: Wii Sports
Wii U: Nintendoland
Switch: Animal Crossing: New Horizons
Gameboy: Tetris
GBA: Pokemon Ruby/Sapphire
DS: Brain Age
3DS: Pokemon Sun/Moon
PS1: Final Fantasy 7
PS2: GTA3
PS3: GTA5
PS4: GTA5 (one reason I think PS4 is a disappointing system)
PSP: Crisis Core: FF7
Vita: No clue. I can't think of one defining Vita game
curl-6 bet me that PS5 + X|S sales would reach 56m before year end 2023 and he was right.
My Bet With curl-6
My Threads:
Master Thread, Game of the Year/Decade
Switch Will Be #1 All Time
Zelda Will Outsell Mario (Achieved)
How Much Will MH Rise sell?
My Bet With Metallox