I would like to remove the name on a photo that was shared with me.

My mistake. I do have iMessage-shared photos with unremovable origin attributions even after I have saved them into my library.

I was thinking of photos shared through Shared Albums. My apologies.

If I duplicate a shared photo, the copy loses the source attribution. Try it and let me know if it works for you.

Is the name a signature on the photo? If the photographer signed the photo with a name it would be bad manners to remove the signature.

That's not a standard feature of shared photos. None of the photos that people have shared with me have names at the top.