HI,
Try these things:-
1) In the Apple Icon Menu top left use the About this Mac option.
In the Overview tab the Serial Number should be visible.
If it is not then visit an Apple Store or Apple Authorise Service Provider (AASP) to have it put back.
It is often missed when repairs to the Logic/Mother board are done.
2) Rest the NVRAM How to Reset NVRAM on your Mac - Apple Support
3) Boot in Safe Mode and them try Messages.
Whether it works or does not reboot normally.
Since OS X 10.8.2 the Messages app has used the Serial Number as part of the Login.
It is part of the process that make s logins device specific.
This means that the iMessages server can lock out the use of the Apple ID on just one device.
Most often this is the Mac.
4) Try logging into a another Mac User Account and setting up Messages with your details.
If this works it is your Users account that has a problem.
If it does not work it is the iMessages Servers.
5) Contact Apple Support by phone.
State you have an Apple ID issue (Free to to fix)
Level one responders are script led and trying to fix Apple Care issues with Software and Hardware.
You are likely to have to ask to speak to a Level 2 or higher assistant.
They still may want to do data gathering and trying some other things.
You do NOT need to Update if at a slightly older version of the current OS.
you do NOT need to Upgrade if using an earlier OS.
You do NOT need to Reinstall.
Insist they check the server and they should be able to see that it is your Mac using the Apple ID that is blocked.
It is likely at this point they will involve the Engineers who will eventually reset the server but you are likely to be told it will take 24 hours.
We have not found a quicker way to achieve this.
9:30 pm Friday; May 6, 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)