Can't pair Apple Watch Series 6 to iPhone 8

Hey there!

1. I'm trying to pair a new Apple Watch Series 6 to my iPhone 8, but it doesn't work.

Steps to reproduce the problem are:

1. Open the Apple Watch App on your iPhone with iOS 16.4 installed
2. Start the pairing process, scan the Apple Watch
3. See that the Apple Watch App informs you that there is a watchOS update available and that you can't continue the setup without updating
4. Tap on "Update now"
5. Get the error "Connection failed" and notice that you are stuck in a loop

Any tips or ideas what I could do to fix that? I already restarted both the Apple Watch as well as the iPhone, which also has the latest iOS update installed. Also, I uninstalled and reinstalled the Apple Watch app.