Hi,
Also use the Apple Icon Menu top Left and the About this Mac item
In the Overview tab of the panel that opens you should see the Serial Number.
Is it showing ? (we don't need to know what it is just whether it is present)
IF you have a test account in System Preferences > Users and Groups then Log into that (Or Enable the Guest option) and login to Messages/iMessages there.
Does this work ?
This will tell you if the issue is Mac wide (more likely to be at the server end) or Mac User Account specific.
Does it work for another Apple ID ?
Report back at this point because if it is Mac wide and Your Apple ID specific the Fix is a detailed set of instructions.

9:22 pm Thursday; September 1, 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)