He specifically asks which games people personally think are iconic to the system. 

Xbox - Halo CE
360 - Halo 3/Reach
One - Sea of thieves

PS1 - FF7
PS2 - Kingdom Hearts
PS3 - LittleBigPlanet
PSP - GTA vice city stories

GB - Pokemon Red/Blue/Yellow
GBC - Pokemon Silver/Gold
GBA - Final Fantasy Tactics Advance
DS - Mario Kart DS

NES - Super Mario bros
SNES - Earthbound (fav game of all time)
N64 - Mario Party
Gamecube - Super Smash bros Melee
Switch - Super Mario Odyssey

Genesis - Sonic 2
Dreamcast - Sonic Adventure

SNES - Chrono Trigger
PS1 - Chrono Trigger
NDS - Chrono Trigger
PC - Chrono Trigger
iOS - Chrono Trigger
Android - Chrono Trigger

I guess I'll do the deep dive to the 2nd gen.  Donkey Kong was the pack-in game for the ColecoVision and was heavily leaned on in marketing to support their "The Arcade Quality Video Game System" tag line.  Donkey Kong helped move a lot of ColecoVision units in a short period of time, breaking Coleco into the home video game market which had been dominated by Atari up to that point.

Ooof, can't believe I forgot Titanfall. Definitely a console defining game. 

N64: Smash Bros.
Gamecube: Melee
Wii: Wii Sports
Switch: Breath of the Wild
Genesis: Sonic
Xbox: Halo
Xbox 360: Gears of War

SNES for me has to be Starfox, as it was the game that came with the system for me, and its spirit of bold ambition and taking the first steps into the 3D realm is a big part of the Super Nintendo's personality for me.

N64 would be Banjo Kazooie, as Rare's output on the system was a huge part of the console's appeal for me, and of those BK was the most quintessential.

Wii, gotta be Super Mario Galaxy, that game's freewheeling sense of innovation and adventure summed up what I loved most about that era of Nintendo.

Wii U, Splatoon. Its the main thing to come out of that generation that felt new and of the time rather than a continuation of past trends.

PS1 will always make me think of Crash Bandicoot; I didn't have a PS1 growing up so I mostly played it when visiting friends and family, and this was the first game on it that hooked me.

PS3 has to be Uncharted 2, it was such a huge event when it came out and finally made the PS3 feel worthwhile.

Xbox 360, probably Gears of War, as the game that first showed what a huge leap it was over the previous generation, and which set the tone for much of the generation that would follow.

NES - Super Mario Bros 3

SNES - Super Mario World

N64 - Super Mario 64

GCN - Super Smash Bros Melee

Wii - Super Mario Galaxy

WiiU - Super Mario 3D World

Switch - The Legend of Zelda Breath of the Wild

Gameboy/Color - Pokemon Red/Blue/Yellow

Gameboy Advance - Golden Sun

Nintendo DS - New Super Mario Bros

Nintendo 3DS - Fire Emblem Awakening

PS1 - Final Fantasy VII

PS2 - Kingdom Hearts or Final Fantasy X

PS3 - Final Fantasy XIII

PS4 - Final Fantasy XV

PSP - Final Fantasy Dissidia

Xbox - Halo

Xbox 360 - Gears of War

Xbox One - Forza Horizon (any of them)

Xbox Series - Halo Infinite

Sega Genesis - Sonic the Hedgehog 1, 2 or 3 and Streets of Rage

Neo-Geo - King of Fighters (any of them) or Metal Slug (any of them)

Steam - Left 4 Dead 2 or Portal 2

NES - Super Mario Bros.

SNES - Super Mario World

Genesis/Mega Drive - Sonic the Hedgehog 3 & Knuckles

Saturn - NiGHTS: Into Dreams

PS1 - Final Fantasy VII

N64 - Ocarina of Time

Dreamcast - Sonic Adventure

PS2 - GTA: San Andreas

Xbox - Halo

GCN - Super Smash Bros. Melee

360 - Halo 3

PS3 - The Last of Us Part 1

Wii - Wii Sports

Wii U - Splatoon 1

PS4 - Horizon: Zero Dawn

XBO - Halo 5

NSW - Breath of the Wild

PS5 and XBSX - Too early to tell.
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Game Boy - Pokemon Red/Blue/Green/Yellow

Game Boy Color - Pokemon Gold/Silver/Crystal

Game Boy Advance - Metroid Fusion

DS - New Super Mario Bros.

3DS - Fire Emblem: Awakening

what about GTAV PS5 version?