At first, sure. The timing of it was perfect and there were quite a few Sega games that were exclusive to the original Xbox (and still are, grumble grumble). Although PS2 and GameCube did get their fair share of Sega exclusives too like Shinobi, Virtua Fighter 4, Super Monkey Ball, and the Sonic Adventure ports. That all changed during the PS3/360/Wii era when Sega released games that felt like they moved on from their Dreamcast "roots." Before, it felt as if the PS2/Xbox/GameCube games would be possible on Dreamcast, but when we got to the PS3/360/Wii era, we were seeing a new era of Sega as well. And they were willing to put their games out on as many systems as possible. I feel like Sega expanded on this, and they even now put out a lot of games on PC too. In fact, currently, I think Sega is more friendly with Sony and Nintendo by giving them exclusives like Yakuza and certain Sonic games (for the Wii U at least) than they are with Microsoft at the moment, but I guess that could be said about many publishers right now.
I will say, though, that it still feels better to play Sonic games on Xbox than PlayStation, even to this day. So maybe that has something to do with it.