Hi,
If you have tried and NVRAM reset and a Safe Boot plus tried the same ID in another Mac User on your computer AND the Serial Number can be seen then the only recourse is to contact Apple Support (Apple Care).
The Level 1 responders are Script led and will try to charge if you don't have Apple Care that is current.
However Apple ID issues are free to fix and this happens to be an Apple ID issue.
The Mac uses the Apple Serial Number to create and device specific Login.
If the servers don't like this for any reason they will lock the device out of the server.
It will only be the iMessages server AND only for one device.
The Level 1s will still try and get you to do things on their Script.
Ask to speak to someone higher.
Some Level 2s do know about this issue not all do.
At OS X 10.8.2 they also used to reset the iMessages server if you asked them.
This no longer happens and if they do involve the Engineers early on in your conversation it is likely you will still be asked to try things and do data grabbing exercises. (Apple know it happens but not why as it seems to effect people randomly)
Insist that the server is reset.
Judging by other people who post back about their experiences, you can be told a reset can take between 24 hours and 5 days.
There is no "Fix" within the app and Reinstalling, Upgrading or just Updating will no t fix the issue as it is the Server end.
7:11 pm Sunday; July 17, 2016
iMac 2.5Ghz i5 2011 (El Capitan)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
iPhone and an iPad (2)