"Add Faces" greyed-out; can't use Faces & Pets.

Where are you seeing the greyed-out "Add Faces"?

It is normal, that the People&Pets album is initially nearly empty, when you create a new Photos Library.

On a Mac you add more people to the People&Pets album by finding some photos of the people or Pets you want to add, where where Photos could detect a face. Enable the option "View > Show Face Names". in the View menu. View the photo enlarged and if you can see a circled face of a person you want to add the People album, type a name into the text field below the face. This will add the person to the People album.

Only for photos, where Photos could not detect any face, is it necessary add a faces circle manually. In that case we can use the "(+) Add Faces" in the Info for a photo, when we open the photo to view it enlarged. This button is only available in Photos for Mac, not on an iPhone or iPad. Click the circled (+), not the grey text "Add Faces".

Could you clarify-- when you say iCloud Photos, to you mean what you see using a browser at iCloud.com? Or do you mean you have a Library on your Mac named iCloud?

I have maybe ten Photos Libraries, and I pop back and forth with them all the time. No problem.

If you have pictures that were added to your phone by some process other than syncing with Photos, then they are not part of the Photos Library and you can't work with them in the same way and Library pictures. These pictures are stored in Files, and they appear in a folder called "From My Mac." If that's what's going on, you need to import those pictures to Mac Photos and then sync them with your phone.

On my iPhone I am only seeing one suggested face in the "Add People" section at the bottom of the People & Pets album, and that is not even a face, just a part of a map. But the "Add" button will become active, after I select the face.