Platformer - Super Mario Bros. 3, mostly for the time it came out and how much that influenced me, the SMW games are also right up there
Puzzle - Tetris, really can't do better than the classic
RPG - Final Fantasy VI
Action - The Legend of Zelda: A Link to the Past, I consider this in the "Action" category which is a super broad category!
Fighting - Super Smash Bros. Melee, despite never owning a GC, I felt this was the best one
MMO - Final Fantasy XI
FPS - Goldeneye 007, what a defining game!
Rhythm - Theatrhythm Final Fantasy Curtain Call
Adventure - Zero Escape: Virtue's Last Reward, I'm considering the old school definition of the point and click "Adventure" category
Racing - Mario Kart 64
Sports - Super Punch Out!!
Mobile - Pokemon Go
Survival Horror - N/A, don't really play these
4X - Age of Empires 2, probably the only one I've played
Tactical/Strategy - Final Fantasy Tactics
Sandbox - Pirates! Gold, I'm just going to call this the Simulation category
Visual Novel - OK I just saw this so I guess ZE:VLR goes here, though I'm not sure what "Adventure" is?
Stealth - Metal Gear Solid 3: Snake Eater, curious category since a lot of action games include stealth

EDIT: How funny, about half of us considered "Adventure" as the old point and click variety (but then what is a Virtual Novel?) and half considered it "Action/Adventure" like Zelda.

Also you need a Metroidvania category, WK. 2 of my top 10 games are in that category (and defined it).