How do I add photos to a pet/person

Only Photos on a Mac has a tool to mark custom faces, that Photos has not detected on its own.

If you have a Mac as well and are syncing your devices with iCloud Photos you could add the missing face on your Mac and it will sync to your iPhone.

As a work-around, if Photos does not recognize the cat as a pet, type the name into the description field below the photo. then a search for the name will find this photo too. this will work in the most recent versions of iOS, not in iOS 10 as shown below your question.

Add someone to the People & Pets album

1. Open a photo of the person or pet that you want to add.
2. Tap the Info button , then tap the face of the person or pet with a question mark.
3. Tap Name This Person or Name This Pet and type their name or tap the name if it shows up.
4. Tap Next, then tap Done.

If you just added the picture, it may take awhile before it's scanned.

So, you see the little icon in the lower left of the picture? What happens when you tap on it?

Use the People & Pets album in Photos on your iPhone, iPad, or iPod touch - Apple Support

The instructions don't say anything about an "add" button. The little button with the face is in the lower-left corner of the picture in the Info panel.

What model phone and version of iOS are you using?

Look at the picture in the article linked to above (shows the name of a black cat being added). When you tap on the little icon in the lower left, does the same menu pop up?

it wont work for me cause when i press the info button now it just searches up what animal it is (species n stuff)