I’m going through the same issue with the Apple Watch series 8. The Watch app crashes at the Share Settings screen. On occasion it says you need to verify your Apple details, but after entering the password it eventually times out saying cannot verify at this time. If you choose forgot password you then go through a password reset which appears to get you passed this error.
The watch also unpairs as soon as the Watch app crashes. This restarts the Watch in the process.
I have done the following troubleshooting steps:
iOS update to 16.0.3
updated Watch to 9.0.3 (I cannot confirm this as a Google search is not showing an updated result. The watch shipped with 9.0.2 and forced me to update before trying to pair)
Force closed app
removed Watch app, restarted phone, reinstalled Watch app
Signed out of iCloud on both my iPhone and iPad, then signed back in
Attempted to pair the watch without restoring a back
Erased all Settings and Data of iPhone and restored from backup
Erased all Settings and Data of iPhone and did not restore from backup
Force rebooted the watch
Getting rather frustrated as I purchased the series 8 to replace my 18 month old series 6 that is stuck in a boot loop.