Caller ID from iPad message shows "Home"
It's not a "Caller ID" as it's not visible to anyone but you.

It's only shown on your iPhone because the phone number (or Apple ID) is in the Contacts of the device with the name set to Home. This means that messages received show the name of the sender pulled from Contacts, just like any other contact on the phone.

If you don't want this, delete the contact from your phone which contains the phone number, but this will delete it from all iCloud synchronised devices and can cause problems if it is your 'Me card' such as incorrect AutoFill information in Safari.