Is it necessary to relogin (logout and login again) in apple services on MacOS after Apple ID password has been changed?
It's not necessary but it's a good practice. Some services will not work because they can no longer log-in.
The services you use for iCloud, iTunes, etc, will be trying to log-in using an old password. That will fail and it will prompt you for your password. This will keep happening until you enter the correct password.
If you want to bypass all those notifications you can log-out and then log-in again with your updated credentials.