Have you also tested in another user account, if the problem is isolated to your current user account or system wide? Log into another user account as described here: How to test an issue in another user account on your Mac - Apple Support
- Launch Photos on a new, small test library and check, if the "Photos" item is there.
If Photos is working on anther user account as it should, the Photos Preferences file may be corrupted.
and removing the following folder may help: ~/Library/Containers/com.apple.Photos/
To delete the Photos preferences files:
- Quit Photos, if it is running.
- Open your user library in your home folder:
Delete the complete folder ~/Library/Containers/com.apple.Photos/ and restart the Mac, then try again. Your user library may still be hidden.
- To reveal the hidden User Library:
- Bring the Finder forward by clicking on the Desktop or the Finder icon in the Dock.
- Select your user/home folder (with the house icon in the Finder sidebar)
- With that Finder window as the front most window, press the key combination ⌘J to bring up the View options.
- In the View options panel enable ’Show Library Folder’. That will make your user library folder visible in your Home folder, if it is still hidden.
- Open your Home folder, then the Library folder, then Containers.
- Move the complete folder ~/Library/Containers/com.apple.Photos to your desktop. Move the entire folder, not just the contents.
- While you are about it, delete also ~/Library/Preferences/com.apple.Photos.plist
Restart the Mac and try again.
After you deleted the ~/Library/Containers/com.apple.Photos/ and preferences, Photos will not remember the location of the library and the preferences you set. You will have to select the library in your Pictures folder and double click it to launch Photos on your library.