Gamecube is the only current-gen console I've purchased after Dreamcast. I still find myself buying and playing more Dreamcast games though.
It's true there is a lot in common between Dreamcast and Xbox, but in the end, it matters little as the types of games each consoles cater to is very different. Microsoft's involvement in developing an OS for Dreamcast may have encouraged them to enter the console business, taking Dreamcast as a benchmark model, but so was Sony. They were impressed by the potential of Sega CD which catapulted them to create a CD-ROM add-on for SNES, which once again catapulted to creating their own Playstation. I see little similarities between SegaCD and Playstation in terms of gaming experience however. Sega's very own SegaCD was inspired from Turbo Grafx CD, but once again, both NEC and Sega went separate ways on contents on each of their CD-ROM add-ons.
Dreamcast may have been a spiritual successor of Saturn in terms of hardware features, (eg. Netlink to DC online) but each consoles offered distinct gaming experience to set them apart. Dreamcast didn't quite rely on PC developers, Xbox took them to the full steam. Xbox is also absent of large libraries of arcade games that made Sega's consoles unique.
The controllers may feel similar, they may share same dedication to online games, they may both show great support for progressive scan, but they are no longer the same when it comes to games. I once felt a sense of apathy towards Xbox when JSRF, Gun Valkyrie, and Panzer Dragoon were announced, but today, I just accept Xbox as a distinct platform of its own.
With the unfortunate demise of DC, I don't feel any sense of belonging towards any of the venders anymore. I'm now impartial towards all three remaining venders. Nintendo and Sony may have damaged Sega in the past, but it's all in the past.
Very few of my Dreamcast friends see Xbox as a spiritual successor of Dreamcast, like me. Most think Xbox as a heavy derivation of PC, having some features of DC intact, but not enough to feel an attachement towards it.