Sort of. DirectX was an entire collection of programming applications, not the application itself. Basically, the X was a stand-in for all the various Direct APIs that made up the suite. DirectDraw was the window one you’re probably thinking of. There was also Direct3D, DirectSound, DirectMusic, and DirectPlay.
Another way to think about it is that it’s akin to AdobeX. There’s Adobe Photoshop, Adobe Lightroom, Adobe Acrobat, etc…
That being said, it was 100% the basis for the Xbox’s name. It was a DirectX Box, and DirectX was (at least at first) the primary method of writing games for the console. And since Windows also used DirectX, it made games much easier to port to PC. By that point, the X had sort of taken on a life of its own, and Microsoft started using it simply as a way to signify that something was made for the Xbox. Xinput, for instance, is the protocol that Xbox controllers use. But the X doesn’t stand for anything in that case, except to signify that it was designed for use with the Xbox.