Hello steveytee,
Are you signed into the same iCloud account on both devices? If so, is the Messages in iCloud feature enabled? You can check on that using the steps below:
1. Go to the Settings app.
2. Tap your name.
3. Tap iCloud.
You can also find those steps in this resource: Use Messages in iCloud - Apple Support That feature keeps your messages synced between all your devices, and could be related to the behavior you are experiencing.
If you aren't using that feature, do you have Text Message Forwarding enabled? How to forward SMS/MMS text messages from your iPhone to your iPad, iPod touch or Mac - Apple Support
Should you still continue to get messages on both devices, you'll want to reach out to Apple Support so they can look into this further: Contact - Official Apple Support
Take care.