It doesn’t work recognition people and pets in iPhone Photos

I would refrain from turning iCloud on and off, if your Photos Library is large. That will just the Photos library to be uploaded agin to iCloud and merged into the existing library, and thecae recognition and other processes will have to start over again. You will just have to wait many days longer until the detected faces will appear.

On iOS 18 Photos is hiding some of the detected faces, until we add them and does nit even show the tiny faces icon in the info. Have you already looked at the bottom of the People&Pets collection, if there is a the option "Add People" shown at the bottom of the album? See: Find and name people and pets in Photos on iPhone - Apple Support (CA)

If the person or pet you want to name doesn’t appear, tap Add People at the bottom of People & Pets to see more faces.

I am seeing to people to add, when I click there. I wonder where he second face is hiding.

It seems to be intentional, that Photos is now showing less faces to name than before, on the iPhone as well as on the Mac. previously I could also name the faces of artwork, like statues or paintings, but that is no longer possible.

Do you have at least 10% of your memory free on our phone?

Have you tried a forced restart? 

Force restart iPhone - Apple Support

This is different from a simple Power Off. I had to continue holding the button down past the "slide to power off" message, and then I released the button only after  the Apple Logo appeared indicating a re-start. 

Do the people and pets still show up in and get added into the People and Pets album?

Which version of iOS 18 do you have installed? Is there an update available for you?