Family's text notifications showing up on my lock screen

I dug around yesterday after responding to your question and was only able to find a couple of old posts from here on the ASC. They didn't seem to be quite the same as yours, and did not offer any resolution. Hopefully if you chat with support they will be able to help. Good luck.

I've been using Family Sharing since the beginning of it, but have never witnessed this behavior. Do you have Parental controls setup for people in the Family group? I cannot seem to find a definition anywhere explaining it. Maybe someone else will read and respond to this that may have an idea. If not, you can also try Apple Support. You can try a chat with them by using the Apple Support app. Download that to your device and use it to contact Support.