Using an Apple Watch with Luna Display

An Apple Watch may not be able to unlock your Mac if the Mac detects any screen sharing software is in use. You may need to quit these apps, Luna Display included, before Auto Unlock can work properly. It is recommended to disconnect from Luna Display if you frequently rely on using your Apple Watch to unlock your Mac.

From the Apple support article on using an Apple Watch to unlock a Mac, there's a short bullet point about Internet or screen sharing potentially causing difficulty for the Auto Unlock feature:

Since Luna Display uses some of Apple's public APIs, the Screen Recording permission is needed to transmit information from your Mac to your secondary device, although we do not record the contents of your screen or store data anywhere. We are not able, and do not wish to, bypass Apple's built-in security features, so this is not something that can be adjusted on our end.