Hi,
You have actually spoken to Apple and they have confirmed that the use of your Apple ID to the iMessages servers from your Mac is not the cause ?
To be clear the Serial Number (or rather lack of it) make the Login Device Specific.
As the iMessages are a standalone set (farm) then it means the use of the Apple ID for the iMessages account is also App Specific. This means the ID will work in other apps and services. It will also work on other devices.
7:28 pm Sunday; November 19, 2017
iMac 2.5Ghz i5 2011 (Sierra)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
iPhone 6 iOS 11.x and an iPad (2)