You can only name faces, if Photos has recognised the face in the photo. If iOS 14 is installed on your device, you could add the additional names to the caption of the photo. Then a search for this name will also find the photo.
See: View photos and videos on iPhone - Apple Support
"Captions add context to your photos and videos, and you can search for photos and videos by captions when you use Search. Select a photo or video, then swipe up to view a caption, or add a caption in the text field below the image.
When you swipe up on a photo or video, you also see the following details:
In Shared Albums, you can add comments and likes to photos and videos. Your comments and likes are shared with the album subscribers. See Share iPhone photos with Shared Albums in iCloud. "