From what I could find, the Series X Xbox that OP has may have a wireless earphone connection ability, but it is NOT standard Bluetooth - it uses a different frequency. So you CAN buy an adapter for that Xbox to give it standard Bluetooth ability and then use any standard Bluetooth headset with that. One small note there - the ones I looked at all have a mic built into the adapter, and it can NOT let you use any mic on your headset. You MUST use the mic in the adapter if you need one at all.
OP, NOTE that this ability to pair two sources to one Bluetooth headset requires the latest Bluetooth 5 AND requires that the headset include a feature called Multipoint or Dual Pairing. This web page shows many details
Bluetooth multipoint lets you connect two devices at once to your headphones. It can greatly enhance your experience, plus it's easy to use.
www.soundguys.com
And this one is by a specific headset maker
https://govo.life/blog/do-more-with-dual-pairing-bluetooth-headsets
So, that's an option for you. IF your computer has Bluetooth 5 built in (OR maybe you can add that with an adapter or PCIe card) AND you buy an adapter for your Xbox Series X AND you buy a headset (with or without mic) that HAS the Multipoint feature, then you can pair that headset to BOTH Bluetooth sources and switch between them with a button on the headset.
However, on your COMPUTER you will need to make a choice, I THINK. Under Windows, no natter how many devices you have installed that can send out sound, Windows can only use one of these at a time. On your screen at bottom left type in Sound Settings to open a window. At the top there's a drop-down chooser that will show you all the possible sound output devices you have. Right now you likely will see a mobo system by Realtek, another named like your video card, and maybe another? YOU get to tell Windows which way to send out sound, and now I would bet it is set to use the video card output system to feed via the HDMI cable. If your system also has Bluetooth ability built in, you may see something like that to feed sound to s speaker set. If not, and you ADD a Bluetooth adapter in some form, then a new Sound Output Device may appear here. Then you have to tell Windows in this screen to use that output device instead of the one on the video card. At a later time you can come back here and change that setting.
There are SOME sound output devices that come with device drivers that allow sound to be sent out from TWO different outputs even though Windows itself does not do this, so read the instructions with whatever you get carefully. In fact, it would be good if you can read such instructions BEFORE buying particular units, OR talk to their respective Tech Support people to be SURE their units will do what you want.