Sega started porting many of their new arcade games to the Xbox. Consequently former Dreamcast users who wanted to play the new arcade games at home bought the Xbox.
A Sega exec tried to push Microsoft to make the Xbox backwards compatible with the Dreamcast even [1], which Microsoft refused mostly because they didn't want to support online infrastructure for titles that used it. The original Xbox seemed to have been considered to be the favored successor to the Dreamcast by Sega at least at one point.
Then again, Sega's main initial portwork was focused on the GameCube. The Dreamcast games Sonic Adventure 2, Ikaruga, Crazy Taxi, Phantasy Star Online and Skies of Arcadia were ported to the GameCube and not to the Xbox. They also brought originally-for-Dreamcast titles Super Monkey Ball, Beach Spikers and Billy Hatcher and the Giant Egg to the GameCube exclusively.
But it really was. Sega and Microsoft worked closely together or why do you think they implented the Windows CE option? Also all Sega arcades afterwards shipped with Xbox hardware.
Windows CE was ported to a lot of processor architectures at that time. Its architecture is quite different from the Windows NT line. To quote Wikipedia on this
"Unlike Windows Embedded Standard, which is based on Windows NT, Windows Embedded Compact uses a different hybrid kernel."
I am also pretty sure that Windows CE's architecture is quite different from the OS running on the XBox, but cannot give a quotable source on this. At least for the Xbox 360 and XBox One, I am pretty sure that I have read that their kernel is actually not so dissimilar from the kernel used in the NT line. The difference between the Windows NT line and the XBox 360/One OS rather lies in the software running above the kernel.
You can quote me, as I've gotten down and dirty with the Xbox ( https://github.com/monocasa/xbvm ), and have written board support packages for CE for work.
The OG Xbox kernel is an extremely heavily stripped down and modified Win2k kernel. No support for user mode, multiple address spaces, or more than one running process. No win32 in the kernel. USB, sound, and the vast majority of the graphics driver are statically linked into the process executable.
There was a rudimentary os when you boot without gd-rom. It allowed you to set settings, manage memory cards etc I suspect it was loaded just in case where there were no os on the gdrom inserted otherwise the system will boot on the gdrom os. Edit : video link https://m.youtube.com/watch?v=5L9p_Bq_hFk
In what sense? The architecture between these two consoles is IMHO quite different.