Apple Watch Ultra 2 does not pair with iPhone13 with iOS 17.6.1

That is to be expected. The Apple Watch Ultra 2 probably uses watchOS 11 which requires an iPhone using iOS 18. Upgrade your iPhone to iOS 18 before pairing your watch.

In fairness to you, this is an easy mistake to make because everywhere I looked it says watch series 10 pairs with a device running iOS17 or higher which is not the case. I made the same mistake and thought I’d blown hundreds of pounds on a watch I couldn’t use until I looked further into it.