Here is an iMessage checklist. Note that most of the Settings are on the iPhone, but also it is crucial that the iPhone phone number be specified in the Mac's Settings as indicated below. All steps below must be followed:
iPhone and Mac both logged in to same Apple ID. Try logging off and logging on if necessary.
iPhone Settings => Messages shows iMessage as "on" (sometimes toggling this off and then on fixes some things)
iPhone Settings => Messages shows Send & Receive showing both phone number and email (Apple ID)
iPhone Settings => Messages shows Text Message Forwarding and your Mac is listed and shown as "on" there.
iPhone Settings => Messages shows Send as SMS "on" and MMS Messaging and Group Messaging both "on"
On the Mac: Messages => Settings => iMessage tab shows same and correct Apple ID and "You can be reached for messages at" indicates both phone number and Apple ID (email), and "start new conversations from" indicates the phone number. If the phone number does not show up properly on the Mac here, sign out of iMessage on the Mac, then sign back in using the same Apple ID as you use on your phone.
* Don't see the Text Message Forwarding setting on your iPhone? Go to Settings > Messages, turn off iMessage, turn it back on, tap Send & Receive, tap Use Your Apple ID for iMessage, then sign in with the same Apple ID used on your other devices.
Forward text messages from your iPhone to other devices - Apple Support