HI,
Open the About this Mac item from the Apple Icon Menu top left.
In the Overview tab the Serial Number should show.
If it does not iMessages will not work (the reset of the app does just not that account)
Get back to the repairer and have it put back on the logic board (it's supposed to be part of the process)
If one is there check it matches what is on your Mac (Back, Bottom or Battery Compartment)
If it is "wrong" then get it changed.
There is a chance that this has also blocked the Apple ID's use for iMessages by that computer at the iMessages servers.
The Serial Number is used to provide a device specific login.
IF it does not work after correcting or getting the Serial Number unto the Logic Board you will have to contact Apple Support (Same as Apple Care).
Once you have explained it is an Apple ID issue it should be free to Fix. That does not mean that the Level 1 person you speak to knows what to do (they Are Script Led for Apple care issues)
Politely ask to speak to someone higher.
Even then they may insist on data gathering and hoop jumping.
Based on others who have replied it is now getting the Engineers involved (the issue happens separately from the Logic Board issue)
Insist the Server is reset regarding your Mac's use of the Apple ID.
Updating or Upgrading does not work and neither does a Reinstall.
Even then they may tell you it will take days for a reset to filter through.

8:43 pm Friday; June 10, 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)