Hi,
Unfortunately some people at an Update or Upgrade find the iMessages account will not Login again.
However other than trying to send iMessages there is no way to see if your are logged in.
The only thing you can do is test.
On the Mac go to Messages > Preferences > Accounts > iMessages and reduce the Receive At list to just the Apple ID.
ON the iPhone in Setting > Messages > Send and Receive reduce the active items to just the iPhone Number.
Then on the Mac start New Message
Manually type in the iPhone Number including the Country Code (it will not pull up your My Card info if you try your name as you are not expected to Call Yourself).
Send an iMessage.
If the iPhone does not see it then the Mac version is not logged with the Apple ID (it is most likely it was "seeing" sync "copies" via the iPhone Number).
The reverse (using just the iPhone Number on the Mac and Sending From the Apple ID on the iPhone) is only partial as the iPhone Number is always logged in and cannot be turned Off - meaning the sync link can explain why you might see some iMessages.
Anyhow if the Mac does not appear to be logged in with the Apple ID you could try Signing Out then In again (restart the app in between).
If that does not work then try this Users Tip
When the iMessages account will not Login
The last section is the most important.
After calling Apple Support (It is an Apple ID issue which is free to fix) you need to work up the levels of Responders until one can look at the server info and see if the Server is blocking your Apple ID when used by you Mac.
The Use of the Serial Number makes it Device Specific.
AS the Apple ID can be different to the one used in iCloud or iTunes then it can be App Specific as well.
This means it may only be one of your Macs.
This issue can also be created if you Migrate a whole Mac User Account to a new computer as the Keychain holds a Key called an Auth Token that holds th Serial Number info
7:59 pm Tuesday; January 23, 2018
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)