3D Mario
TLOU
Final Fantasy
Halo
Forza
God of War
Mario Kart
GTA
Dragon Age
Mass Effect

Nintendo: Fire Emblem, Metroid
Sony: Crash Bandicoot, Gran Turismo
Microsoft: Halo, Age of Empires
Third Party: Civilization, Elder Scrolls
Personal: Zelda, Mario

Loved the idea.
My console is called OMEGAPOWER, a cool next gen system.

Two franchises from Nintendo: It has Fzero games and Wrecking Crew games.
Two franchises from Sony: Until Dawn and Siren games.
Two franchises from Microsoft: No, thanks.
Two franchises from third-parties: Civilization and Tropico.
Two flex franchises from whichever categories you want: Silent Hill and GTA.

Two franchises from Nintendo - Donkey Kong Country, Mario Kart
Two franchises from Sony - Uncharted, Ratchet & Clank
Two franchises from Microsoft - Gears of War, Banjo
Two franchises from third-parties - Dark Souls, Fallout
Two flex franchises from whichever categories you want - The Legend of Zelda, Super Mario 3D

Great topic!

1. The Legend of Zelda

2. Super Mario platformers

3. Team Ico franchise

4. Ratchet and Clank

5. Ori and the Blind Forest

6. Halo

7. TimeSplitters

8. Resident Evil

9. Fire Emblem

10. Mario & Luigi

It pains me to leave out the following though: Donkey Kong Country, Metroid, Xenoblade, NiGHTS, Panzer Dragoon, Final Fantasy, Doom, Portal and Metal Gear Solid

I included Games from Sega,Indies and PC:


Two franchises from Nintendo
-Super Mario 3D
-Pokemon

Two franchises from Sony
-The Last Of US
-God Of War

Two franchises from Microsoft
-Halo
-Forza

Two franchises from Sega
-Sonic
-Bayonetta

Two franchises from PC
-Half Life
-Civilization

Two franchises from third-parties
-GTA
-Darksouls

Two franchises from Indies:
-Rocket League
-Shovel Knight

Two flex franchises from whichever categories you want
-Kingdom Hearts
-Super Smash Bros

Dang. That's hard. While I love many Nintendo and third party franchises, I can't really think of many first party Sony and Microsoft games I just need to play.

2 Nintendo franchises-Has to be Mario and Zelda.
2 Sony-Shadow of the Colossus. Then, I guess, Rachet and Clank
2 Microsoft....I'll just say 'the Rare IPs if they can actually do something with them'
2 third party-Star Wars Rogue Squadron, Mass Effect
2 "miscellaneous'- Super Smash Brothers, Fire Emblem

Well, if I'm making my own dream system....

Ratchet & Clank
Sly Cooper
3D Mario
Zelda
Gears of War
Banjo
The Elder Scrolls
Kingdom Hearts
Jak and Daxter
Pokemon

I like platformers...

Zelda & Fire Emblem

Last of US & Uncharted

Fable & Banjo

Elder Scroll & Civilization

Witcher & Ratchet & Clank

Hey, five in common! And I was THIS close to picking Elder Scrolls instead of Street Fighter.