Hi,
1) Using the Apple Icon Menu > About this Mac > Overview tab can you see the Serial Number listed near the bottom ?
2) In Applications/Utilities open Keychain Access and then set the top left list to "Login" and the bottom left to "Passwords" then on the right can up find the IDS: YourAppleID-Auth Token item ? (YourAppleID should be the email you are using for Messages and the iMessages account)
3) Try a NVRAM reset. How to Reset NVRAM on your Mac - Apple Support
Any changes ?
4) Use either the Guest account or create a Mac User Account (System Preferences > Users and Groups) and test messages there using the same details.
Does it work in the other Mac User Account ?
5) Try a Safe Boot
Does Messages work there fro the iMessages account ?
If it does Restart normally.
6) IF 2) is present and 3) and 5) have not helped then delete 2) and restart Messages.
Does the IDS item reappear ?
If none of these work but the Serial Number is present you will need to Contact Apple Support and explain that you have an issue with an Apple ID in relation to your Mac using it in iMessages.
The people who answer the phone are Level 1 responders and tend to be using a script. (they are working to solve Hardware and Software issues for Apple Care users)
Apple ID issues are free to fix.
Ask to speak to someone higher.
Some Level 2 are aware of the problem and may be able to reset the iMessages Server for your ID.
It is the Serial Number that make the use of the Apple ID device specific.
It was added (the use of the Serial Number) as a security feature but it seems in some cases it gets triggered and you then cannot login.
It needs a reset at the iMessages servers.
More commonly recently it has been passed over to the engineers and reset can be as long as 24 hours.
The Engineers are still trying to find out why this happens.
You are likely to be asked to jump through many hoops and data grabbing exercises along the way.
Stop short of doing a Reinstall (or Upgrade/Update to a later version).
Insist, Politely, that the server is reset.
If the Serial Number is missing from your Mac you will need to visit an APple Store and have it put back.
It often gets missed when the Logic Board is replaced.

9:23 pm Tuesday; March 29, 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),
Couple of iPhones and an iPad