How can I add my pet’s name back to the Photo?

If Photos recognizes that there's a face, but it's not sure whose face it is, then in the info window there will be a little face icon with a question mark. Then you can click on that and provide a name. If there's no little face icon, then Photos didn't see a face there, even though you do. In that case you can't add a name to appear in the People & Faces albums.

But you can add the name to the caption in the Info window. The face IDs are cool and all, but they're not universal. For instance, when you export a picture, it doesn't include the face ID. But it does include the caption. So I put names in captions, even for people (and pets) that have been ID'ed, so that I can search for them, and I can depend on the names remaining for exports and after system updates or catastrophes.