Every video game system, whether it be console ot handheld, has its library of games that everyone remembers fondly enjoying on that system. But I believe there is always that ONE game in particular that serves as THE game on the system. For a multitude of reasons: Quality, sales (software AND hardware), franchise or genre defining, pushing the system to its absolute graphical and technical limits, taking full advantage of the system’s unique hardware or gimmicks (Motion Controls, Touch Screen), etc. Basically, if you think of that system, you think of that game.

For me, personally:

NES - Super Mario Bros.

SNES - Super Mario World

Genesis - Sonic the Hedgehog

N64 - Super Mario 64

PS1 - Final Fantasy VII

GB - Pokémon Red/Blue

GBC - Pokemon Gold/Silver

PS2 - GTA: San Andreas

GBA - Pokémon Ruby/Sapphire

OG Xbox - Halo 2

GameCube - Super Smash Bros. Melee

Wii - Wii Sports

DS - New Super Mario Bros.

360 - Halo 3

PS3 - Last of Us

3DS - Pokemon X/Y 

Wii U - Splatoon

XOne - Halo 5

PS4 - Last of Us 2

NSW - Breath of the Wild

Disagree on some of those, although notable games I don't feel some of them were the defining games I'd also say some platform had around two or so of them, * are an additional defining game that are just as regarded while those with out I feel are the more defining game of the platform, another game may be better or sold more but the chosen game I feel set the tone more for the platform:

SNES - Donkey Kong Country*

N64 - Zelda: OOT*

PS2 - GTA3

DC - Shenmue

GC - Metroid Prime*, Resident Evil 4*

XBX - Halo

360 - Gears of War

WiiU - MK8

PSP - Monster Hunter Freedom/Portable

PS Vita - Gravity Rush

For me its....

Sega Genesis: Sonic 3 and Knuckles combined

Sega Saturn: Did not own one, but I assume a Sonic game

Dreamcast: Sonic Adventure

NES: Legend of Zelda

SNES: Chrono Trigger

N64: Ocarina of Time

PS1: Chrono Cross

PS2: GTA 3/Vice City/Dan Andreas

XBox: Halo

GC: Wind Waker

360: Halo 3

PS3: The Last of Us

PS4: God of War

X1: Ummmmmm....

WiiU: Breath of the Wild

Switch: WILL BE Breath of the Wild 2

And for dedicated portable devices...

Gameboy: Link's Awakening

Game Boy Advance: A Link to the Past

DS: Chrono Trigger DS

3DS: A Link Between Worlds

Vita: Persona 4 Golden

PSP: God of War: Ghost of Sparta

PS1: Final Fantasy VII
PS2: GTA: San Andreas
PS3: The Last of Us
PS4: Uncharted 4
PSP: GTA: Liberty City Stories
Vita: Hard to say. Either Uncharted: Golden Abyss or Persona 4 Golden.
NES: Super Mario Bros.
SNES: Super Mario World
N64: Super Mario 64
GCN: Super Smash Bros. Melee
Wii: Wii Sports
Wii U: Mario Kart 8
Switch: Animal Crossing: New Horizons
GB: Pokemon Red/Blue
GBC: Pokemon Gold/Silver
GBA: Pokemon Ruby/Sapphire
DS: Nintendogs
3DS: Mario Kart 7
Xbox: Halo 2
Xbox 360: Halo 3
Xbox One: Halo 5: Guardians
Genesis: Sonic the Hedgehog
Dreamcast: Sonic Adventure
2600: Asteroids to define the console's sucess. Pac-Man to demonstrate the video game crash.

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

NES: Super Mario Bros. The game that set the console off and arguably saved the console market in the US.

SNES: A Link to the Past or Super Metroid. These games are basically considered the pinnacle of 2D gaming and 16-bit design.

Genesis: Sonic the Hedgehog. The game that put Sega on the map.

Saturn: Panzer Dragoon. The first game I think of when it comes to the Saturn.

Game Boy: Pokemon. It's Pokemon, nuff said.

PS1: Either FF7 or Metal Gear Solid. These games personify the move to more cinematic games on the Playstation.

N64: Super Mario 64. It's been called the only launch title that truly showed off what the console could do for a reason. So much of the book for 3D game design was written by Super Mario 64. Shout outs to Mario Kart 64 and Goldeneye for truly pushing the 4-player multiplayer aspect of the console.

Dreamcast: Sonic Adventure. It would be Shenmue if that game came out earlier, but Sonic Adventure was a statement and its orca chase in the first level is one of those generation-defining moments.

PS2: GTA 3. Gaming was getting grittier and more mature, and this was the game that really established that trend.

Xbox: Halo. The game IS Xbox at least for that generation.

Gamecube: Either Melee or Wind Waker. Melee is by far the most popular game on the system, while Wind Waker exemplified a lot of what defined Nintendo in those days, making unexpected and risky changes while largely ignoring the more mature direction the other console manufacturers were going in.

GBA: Pass. I can't think of anything that stands out from the rest.

DS: Mario Kat DS. This game was a phenomenon and everyone was obsessed with it, plus it added online play to the series pretty well. Still considered a high water mark for the series as a whole.

PSP: Ridge Racer. The game that seemed like it was from the future.

360: Gears of War. The game that really showed what the system could do at the perfect time while popularizing a new form of action game and giving people the first real reason to get Xbox live that wasn't Halo 2. Call of Duty would supplant it in influence just a year later,but Gears gets the nod for being first and for being an exclusive.

PS3: Uncharted 2. Everyone's favorite Naughty Dog game except for maybe The Last of Us. I think it also marked something of a turning point for the system's fortunes, as it had been floundering for the first couple of years after launch.

Wii: Wii Sports. It couldn't be anything else.

3DS: Super Mario 3D Land. The game that showed how the stereoscopic 3D should be done and even used it to influence gameplay.

Vita: I can't think of games that came out on the Vita period.

Wii U. Super Mario Maker. A game the system was basically made for.

PS4: Again, hard to think of games that stand out. Maybe Horizon or God of War, but something that defines the system? Don't really see it. Same for XB1.

Switch: Doom. I know a lot of people will go straight for one of the big first-party titles, but to me Doom defines the Switch's mission statement of the same big console experience but on the go. All the impossible ports from Witcher 3 to Crysis are following on from what Doom did.

I didn’t know Brawl was on the GameCube.

Heh, my bad.  I always confuse Brawl and Melee.  I'll edit my post. 

As far as system sellers and popularity the first post looks good, and those games sold a lot of systems. As far as the game that defined the systems to me personally.

NES. Super Mario 3 or Legend of Zelda
SNES. Chrono Trigger. many great games but nothing compared for me.
N64. Goldeneye ,Legend of Zelda Ocarnia of Time, and Mario 64. Impossible to pick just one here. Mario 64 drew me in stunned by the graphic and freedom, Goldeneye was probably my 2nd most played game of all time with all the multiplayer fun with the neighborhood kids, and Zelda was perfect.
PS1. Resident Evil 1 and 2
Gamecube. Resident Evil 7 or Metroid Prime
PS2. Grand Theft Auto Vice City
Wii. Wii Sports or Super Mario Galaxy
PS3. Uncharted 2 or Last of Us
Wii U. Mario Maker
PS4. Horizon or Last of Us 2....Divinity Original Sin 2 is very close
Switch. Hard not to say Animal Crossing..but for me personally Zelda BOTW.
PSVR. Resident Evil 7, Thumper

I will only mention the platforms I know well:

NES: Super Mario Bros. (I think we all agree on this one)
SNES: Street Fighter II
N64: The Legend of Zelda: Ocarina of Time
GB: Tetris
PS1: Final Fantasy VII
GC: The Legend of Zelda: Wind Waker
GBA: Golden Sun
PS2: God of War
Wii: Wii Sports
NDS: Mario Kart DS
PS3: The Last of Us
PSP: Crisis Core: Final Fantasy VII
Wii U: Splatoon
3DS: Animal Crossing: New Leaf
PS4: The Last of Us Part 2
PS Vita: Gravity Rush
Switch: The Legend of Zelda: Breath of the Wild (ATM)