This solution worked for me by Ralph Johns from Messages says "Not delivered" on Apple Support Communities. It seems to be the same problem as described here.
AS a test deselect the iPhone Number as a Receive Item (it will close the Drop Down for Start Conversation From) so that the Mac can only Send (And Receive) from the Apple ID
Now start a New Message.
in the "to" spot type in your iPhone Number in full including the country code.
(The My Details are not called up if you try your name as you are not expected to call yourself).
Before send go to the iPhone > Settings > Messages > Send and Receive and deselect the Apple ID (don't log it out just deselect it).
This bit will stop any Sync copy showing up from the Send of the Apple ID on the Mac.
Now send an iMessages from the Mac.
Does the iPhone Receive it ?
If not then the Apple ID is not logging in properly on the Mac.
As a first Step try Logging out (Sign Out button) on the Mac version. Quit the app before relaunching the app and signing back in again.
Repeat the test above.
If it still does not work then use this Users Tip When the iMessages account will not Login
There is no way to speed up the last section where you have to contact Apple Support and get yourself moved up the chain of Responders until one can look at the Server and tell if your Apple ID when used from the Mac for iMessages is blocked. (your serial number is used for a device specific login so only one device can be block meaning the ID works in other places and devices).
You then still have to wait for the Engineers to reset your ID.