Do the Xbox Series X and Series S use the same power cable?

Yes. Both consoles use the same standard IEC C7 "figure-8" power cable, and that cable is also interchangeable with the PS5, many TVs, Sky Q boxes, and various audio devices. If you've misplaced your power cable, any C7 replacement rated at 10A/250V with a UK three-pin plug will work safely with either console.

Does the Xbox Series X need an HDMI 2.1 cable?

To get the best out of the Series X, yes. The Series X comes with an Ultra High Speed HDMI cable and, if you want to enjoy 4K gaming at 120fps, you'll want a TV that also supports HDMI 2.1. Without HDMI 2.1, the console will still work — you'll just be limited to 4K at 60fps on HDMI 2.0, which is perfectly good for most gaming.