Have you tried doing an NVRAM reset? First, if the feature is still enabled in System Preferences, try DISABLING it. Then, empty the contents of the ~/Library/Sharing/AutoUnlock folder. Then delete the Keychain Entries that you get when searching for AutoUnlock and Auto Unlock in Keychain Access (for any entries to show, enable ’Show Invisible Items’ in the View menu. Then, after deleting those entries, do the NVRAM reset. Then try enabling again. Note that enabling the feature for the first time nearly always fails because MacOS itself also has a tiny little bug in it; enabling it a second time should work. You may also want to unpair and re-pair your watch prior to the NVRAM reset, but it isn’t an absolute necessity, especially given the fact that your other system works with no problems.
In your case, be sure to only carry out this procedure on the Mac that is affected; leave the other Mac untouched.