PS TV: P4 Golden and FF tactics: War of the Lions.
PS2, PS3 and PS4: The Final Fantasy series.

A Super Smash Bros. game is what really makes a Nintendo console unavoidable for me. However, there's not really system sellers for me, I buy consoles for multiple games generally.

I got my PS1 because of FF VII, Crash and Tekken mostly. Every other Playstation console i've owned since then was because of my my faith in them and their amazing exclusives!
As far as handhelds go, I got the Gameboy for Pokemon Red, Gameboy Color for Pokemon Yellow/Gold, Advance for Pokemon Ruby etc.. Now that I think about it all I ever play on Nintendo handhelds are Pokemon games and to a lesser extend Mario/Mario spin-offs. PsP I got for Crisis Core: FF VII and Kingdom Hearts BBS.

Bloodborne PS4
BotW Switch
OoT 3DS
Pikmin 3 WiiU

Single games that were the reason I bought a particular system:
Consoles:
Intellivision - Snafu
ColecoVision - Donkey Kong
NES - Legend of Zelda
Super NES - Super Mario World
N64 - Super Mario 64
Sega Saturn - Dragon Force
Play Station - Final Fantasy VII
Xbox - Halo
Gamecube - Star Wars: Rogue Squadron II: Rogue Leader
PS2 - Grand Theft Auto III
Xbox 360 - Chromehounds
Wii - The Legend of Zelda: Twilight Princess
Wii U - New Super Mario Bros U
Nintendo Switch - The Legend of Zelda: Breath of the Wild

Handhelds:
Gameboy Advance - Golden Sun
DS - Phoenix Wright Ace Attorney
3DS - Super Mario 3D Land

*I bought my Xbox One originally just so I could play Middle Earth: Shadows of Mordor, but I could just as easily have gotten a PS4 for that reason (and in hindsight maybe I should have)

Breath of the Wild in portable form for the Switch...

Twilight Princess HD for the WiiU (do not judge me)

Killzone Shadowfall made me get a PS4, but Bloodborne was the first real "system seller".

I got an X1 just so I could bash it and not be a fanboy...and Sunset Overdrive and Titanfall were fun.

PS3 was Metal Gear Solid 4.

360 was Fable 3

PS2 was Final Fantasy X

PS1 was Chrono Cross

Gamecube was Wind Waker

N64 was Ocarina of Time (I got the system before it though)

Genesis was just a system I had.

SNES was Chrono Trigger.

NES was my first console.

Gameboy was Mario Land

GBA was Golden Sun.

3DS was A Link Between Worlds

VITA was Uncharted Golden Abyss.

Zelda and Fire Emblem are the two franchises I'll buy a system for, which is why I tend to buy Nintendo platforms shortly after launch.

The Tales of franchise used to be included in that small list, but I haven't felt as passionate about that series as I used to be. The SMT/Persona franchise has probably surpassed that as a system seller for me, though I'm not sure I can say it's reached that level yet.

Edit: I see this is more about specific games. I'm too lazy for that anyway....but I'll say BotW was my reason for buying a Switch at launch, despite it being multi-plat.

Let's see...

SNES - Super Mario World
N64 - Super Mario 64
GC - Smash Bros Melee
Dreamcast - Sonic Adventure
PS2 - Dance Dance Revolution
3DS - Fire Emblem Awakening
Wii U - Mario 3D World

As for other systems, I was too young to have one for the NES and Genesis, bought the GB/GBA/DS/Wii/PS3 for games in general, bought the Switch for the hardware, and never owned a PS1 nor any Xbox systems.

I think that covers it..?

The promise of Rareware games sold me on the original Xbox, yeah, well luckily Halo, some Sega and Tecmo games were worth the price of admission.

Vita - Persona 4 Golden
Somehow. Never played a Persona game, watch a trailer or 2 and was somehow sold on an entire console to play it.

3DS - Ocarina of Time 3D
Pure nostalgia, it helped that it played DS games (aka Pokemon)

PS3 - Final Fantasy XIII
No seriously, I didnt know a single other game on PS3 at the time but I needed one for Final Fantasy. FFX and FFXII had that effect on me.