How do I get “message” off my iPad?

Go to your iPhone that are receiving the original messages

Settings > Messages > Text Message Forwarding > Disable devices.

OR

Remove your personal iCloud messages from the iPad

-on the ipad-

Settings > "Username" > iCloud > Messages > Disable

Email Message, Text (SMS) Message - or something else?

If you have multiple devices signed-in to iCloud - and Continuity Services are enabled on those devices - you will see a lot of services and content synchronised.

More information about Continuity and the relevant Cellular services can be found here:

Continuity

https://support.apple.com/en-gb/HT204681

SMS/MMS Messaging

https://support.apple.com/en-us/HT208386

WiFi Calling

https://support.apple.com/en-gb/HT203032