Hi,
Have you had the Logic/Mother board replaced ?
This should have the Serial Number on a readable chip on it. Sometimes it is not put back.
Open the Apple icon Menu top left and About this Mac
In the Overview tab the Serial Number should show up.
Is it there ? (I don't need to know what it is)
In System Preferences > Users and Groups either enable the Guest option or create another Mac User Account for testing.
Log into the account.
Start up Messages and try your Apple ID in the iMessages account.
If it does not work here try and Different Apple ID.
If your ID does not work but another does test the other ID in your normal Mac User Account.
If that works your account is most like blocked at the iMessages Server.
It is a known issue but for Apple the numbers are small.
As a consequence they don't know why this security feature id getting triggered when it should not.
There is also no reset on the Mac end.
The only fix we have found is to contact Apple Support (Apple Care)
Level 1s are Script led and you have an Apple ID issue which are free to fix and are something they don't normally deal with.
Level 2s may still want you to jump through hoops and data gather. (you may have to limit this)
Reinstalling, Updating and Upgrading do not work.
Only a reset of the iMessages server regarding your Mac's use of the Apple ID will work.
They will probably bounce it to the engineers at this point.
You may get a lead time of 24 hours to 5 days (based on what people who have posted back have said)
9:40 pm Tuesday; August 30, 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)