Actually, after a few researches on the forums, I found out a way to solve this issue.
If you turned on the “Allow your Apple Watch to unlock your Mac” feature, but it stopped working, here’s how to fix it:
- Go to Apple menu > System Preferences
- Choose Security & Privacy
- Click General
- Uncheck Allow your Apple Watch to unlock your Mac
Now restart your Mac and Apple Watch. On your Mac, go to the Apple menu and choose Restart. On your Apple Watch, press and hold the long flat button next to the Digital Crown, then swipe Power Off. After your Apple Watch finishes powering down, press the long button again to turn it back on.
Once your Mac and Apple Watch are back up and running, go to the Security & Privacy preferences on your Mac again and turn on Allow your Apple Watch to unlock your Mac.
It worked on my Mac. I hope it will works on your Mac too.