Why is family sharing sending us all the same messages?

Sounds like Messages is still logged in to your Dad's account.

Go to Settings->Messages->Send & Receive and check what Apple ID is logged in there.

Are you signed into his iCloud account on the iPad? There is no way that I can envision that you would be getting his text messages without sharing an Apple ID or an iCloud account.

I speak from some experience. For years my wife, daughter and I all shared on Apple ID for purchases. We never got each other’s text messages or shared any content at all because we didn’t share an iCloud account. I finally set them up with their own Apple ID’s and set up family sharing a few years ago.

Is it your email addrress under "You Can receive Messages to and Reply from" or under Apple ID at the bottom?

Hmm, then I don't see how you could be getting his messages. I'm assuming you mean his text messages (SMS and iMessage).

Family sharing does not share messages, and the only other way to get them would be if you were using the same Apple ID.

Since you are not, I am a bit stumped.