Shouldn't I be prompted to fill in the new password in FaceTime on the Mac? Seems like a security risk no?
FaceTime and Messages have separate passwords from your Apple ID.
I was referring to this: https://support.apple.com/en-ca/HT204408
Shouldn't I be prompted to fill in the new password in FaceTime on the Mac? Seems like a security risk no?
Not really, your AppleID is immediately secured with the new password but the nature of the communications facilities such as Facetime and Messages is that they need to be able to disconnect and reconnect as your data connection changes, if every temporary disconnection caused you to have to re-enter passwords everywhere that would be quite painful. If you are worried, setup app-specific passwords, then those secondary services aren't loaded with your AppleID password anyway.
If you lost your Mac and were worried about someone seeing your messages and getting your calls (you have an account password to prevent that though right?), then you have options via iCloud to secure your Mac if it comes onto the grid...