Hi,
I have never seen this be an Hardware problem (other than the fact replacement Logic/mother boards have not had the Serial number put back)
You needs to make some checks most of which are in this Thread.
IS the Serial Number present ? (Can it be seen in About this Mac or System Profile or Info depending on OS X version)
Have you tried the same details in another Mac User account or the Guest account ? (See System Preferences > User and Groups).
Try a NVRAM reset
How to Reset NVRAM on your Mac - Apple Support
Try a Safe Boot
Try safe mode if your Mac doesn't finish starting up - Apple Support (this title is only one use for this)
After those checks the only Fix is to Contact Apple Support (Same as Apple Care)
Level 1 responders use Scripts and you need to explain you have an Apple ID issue.
Even then you may have to resist them trying to get you to make checks and try different things.
Ask to speak to someone higher.
Level 2s can be a mix bag (judging by other peoples posts)
Some know about the issue and some don't.
Apple know about the issue but not what triggers this security feature when it should not be.
Ask them to check the iMessages server is not blocking your Apple ID being used by your Mac ?
The Serial Number makes a Device Specific Login - hence why only the Mac is affected.
The test with another Mac User account proves it is Mac wide and not just your Mac User account.
The Level 2 is likely to want to try things and also Data gather.
Reinstalling, Upgrading and Updating will not work - hence asking them to check the server.
The Issue is at the server end.
Ask for a reset.
At this point they are likely to involve the Engineers.
There seems to be two reasons for this - it seems that they are the only ones who can do a reset and they are still interested in grabbing as much data as they can.
8:08 pm Tuesday; August 2, 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)