I had PS2 and GC, but not Xbox. In regards to Sega, the best thing about them going multiplat was being able to play Virtua Fighter 4 on a system that I already owned. VF4 and Evolution blew PS2-era Tekken out of the water. They were the best fighting games of the era.

I prefer Xbox. Jet Set Radio Future became one of my fave SEGA games ever and has one of the best OST's ever.  I also loved how you could burn CD's and games could have custom soundtracks. S controller was the best controller that gen. Of course I also got it for Shenmue II stayed for JSRF and Otogi 1-2. Panzer Dragoon Orta. Gunvalkyrie. Ninja Gaiden. Phantom Dust. I wish I could list Metal Wolf Chaos but it was stuck in Japan so I got to play it on PS4. Plus Xbox at the time felt like the closest thing to a continuation of Dreamcast for me. I like a number of games on Gamecube but dislike some other ones. Never liked that controller. Great hardware tho and a cool design. Just Xbox won me over. I did not have a PS2 until many years later.

I've never had either, but as a PC guy, I feel like this has to go to GameCube. I don't think there was that much going on for Xbox if you had a PC, and that was back when the difference between PC and console versions could be even more significant than today (and that's not to downplay the advantage PC has over consoles today).

Halo 1-2.
KOTOR.
Half-life 2.
Ninja Gaiden Black.
Jade Empire.
Fable.
Max payne.
Chronicles of Riddick.

Xbox wins.
Imho, easy win.

JSRF's soundtrack is burned into my brain. I don't own an OG Xbox right now because finding a working one in good condition is brutally hard. But once I get one I can't wait to jump back into JSRF just for the crazy soundtrack alone! Also, excellent picks overall for OG Xbox games!

Depends on your preferences honestly. For me, it's the GameCube, but the Xbox was cool, too.

I only had a PS2 so I missed out on Sonic Adventure 1 & 2, Sonic Gems Collection, Crazy Taxi 3, Otogi 1 & 2, Phantasy Star Online, Skies of Arcadia, etc.

I think they were the only Japanese 3rd party publisher not afraid to fully support Xbox and Nintendo during the 6th generation, whereas other publishers clearly had a favorite which was obviously PS2.

For me, easily the Gamecube but it's also one of my favorite home consoles of all time. Metroid Prime, Wind Waker, Twilight Princess, REmake, Eternal Darkness, F-Zero GX, Animal Crossing, Rogue Leader and Smash Melee were all fantastic and were all only on Gamecube. My favorite games for the Xbox and the PS2 were Splinter Cell and Resident Evil 4, respectively... and both were available on the Cube. The Thousand Year Door, Viewtiful Joe, Pikmin 1&2, Soul Calibur 2, Ikaruga, Super Monkey Ball, Sands of Time and Capcom vs SNK 2 were also titles I really liked.

Other than needing some play time with Halo and God of War 1&2, I was perfectly content with just owning Nintendo's console that generation - something I couldn't say since the NES or again until the Switch. The only real disappointment on Gamecube for me was, unfortunately, Mario; I didn't care for Sunshine or Double Dash.

Capcom was split between Sony and Nintendo, though generally favoring Sony for that sweet installbase. One obvious weakness the PS2 had relative to PS1 was that it kinda sucked at Resident Evil. Outbreak were cool spinoffs but weird and experimental. Viewtiful Joe, Code Veronica, RE4 were late and mostly inferior ports. Dead Aim sucked. Dino Crisis 3 and P.N.03 were exclusive to Xbox and GameCube (but they were terrible, probably the worst games I played on these consoles). I think Shinji Mikami had some sort of grudge against Playstation's management at the time.

PS2 was the default because of its huge install base, which was almost four times the combined install base of the GC and Xbox. Regardless, Capcom did support Gamecube fairly strongly, though their exclusivity deal, out of necessity, turned into limited exclusivity because of the extremely wide gulf between the PS2 and Gamecube userbases. RE4 on Gamecube alone wouldn't have made enough money to make back the development costs. Namco also made solid games for the Gamecube, which was a first for them since the NES, and in the US, at least, Tales of Symphonia stayed a Gamecube exclusive for almost a decade. 

Sega was trying to find their way as a third party company, and I honestly wish they had more financial success in that regard. I always preferred Virtua Fighter over Tekken, and they have been second to Square Enix as a RPG developer in Japan. Skies of Arcadia Legends was one of the first Gamecube games I bought. VF4 was the first PS2 game I bought after FFX.