I'm guessing "it just prompts me" means in the info window. Here's a picture of my cat, Flurry, who has just claimed a box from Amazon to be her own:

It says "Look up Japanese Bobtail," obviously never having been awakened by that unbobbed tail in the morning. The name Flurry is shown when I click the little circled cat in the corner.
This one, with another box, says "Look up Cat" -- less specific, but more accurate.

Neither one mentions Calico or American Shorthair.
OK, I'm just showing you pictures of my cat. But what do you mean by "it works fine?"
You say, "it removes the name and the ability to add one." I guess you mean that the little circle isn't there, is that right? Well, each device has its own processor and its own algorithms to recognize faces, and it's fairly common to have different devices miss some faces that others find.
If you've had the picture in your iPad for several days, then it's through with the scan and it won't ever re-scan it. You could delete the picture and re-import it from the phone that took the picture, and then it will be re-scanned--but it may still not recognize the cat as a face. It's in the nature of machines to do the same thing every time.
What I should point out is that not all the pictures of Flurry are recognized. Notice in the picture that

I have included the name Flurry in the caption. The caption responds very well to searches so, when I think about it, I try to include the name of people and pets in the caption, both for those that have been recognized and of for those that haven't. When you share a picture, the Face ID is never included, but the caption can be included in the metadata of the picture. With that, a search for "Flurry" should reveal all her pictures.