It happens to me occasionally, and I just helped someone out with it over on Reddit. I figured it may help someone at some point if I also posted it here.
Problem: When receiving a text or phone call, a phone number shows instead of a contact name. Remains true while looking through conversation threads via Apple Watch Messages app.
Solution: Go to the Apple Watch app on your phone > General > Reset > Reset sync data. This will just remove the connection between your iPhone and Apple Watch, therefore removing contacts from the watch! (Note: no contacts on your iPhone will be manipulated/lost) No data will be lost, settings on the device should remain the same. It will take you maybe 15 minutes to reconfigure the connection between the two, and then you should be good to go!
If this doesn't work, feel free to comment and I'll be happy to assist in any way that I can.
