I could get it to work:
I noticed that unlocking the Mac with my Apple Watch didn't work either, so it wasn't just 1Password.
I tried disabling Apple Watch unlock in System Settings, rebooting and enabling it again. This didn't help.
Then I tried restarting the iPhone and Apple Watch. That didn't help at first.
Then I disabled Apple Watch unlock in System Settings again and enabled it.
Now I could use the Apple Watch to unlock my Mac!
Then I tried it in 1Password and it started to work there, too!
So, success, I guess. Not sure which step made it work in the end. I hope it keeps working.
