I have Macbook Air 13 and when trying to access my "Photos" App it has a faded line mark. And says, "You can’t use this version of the application “Photos” with this version of macOS." It says I have Version 5.0

It sounds like you have an old version of the Photos app installed on your MacBook Air – one that is incompatible with macOS 13.5.1 Ventura. The Photos that came with Ventura 13.1 is Photos Version 8.0 (520.0.240), indicating that your version of Photos should be at least that new.

Photos doesn't show up in the App Store, but this Support article suggests that you might be able to download an update using Software Update in System Settings.

Get the latest version of Photos for Mac - Apple Support

If your photos are in iCloud Photos, then I'd assume that you

• Can view them on your iPhone – if it's using iCloud Photos
• Can view them using the iCloud Photos Web interface ( https://www.icloud.com/photos/ )

If you made a copy of your Photos Library, and went through it, I think you'd find that your individual photos were in familiar formats like .JPG and .HEIC. However, the directory structure might be arranged in a way that happened to be convenient for the Photos code, but not for someone trying to manually sort through the files.

With the Mac version of Photos, you can have multiple Photos Libraries. Hold down the Option key when launching Photos to get the dialog for creating and selecting libraries. You can have one "system" Photos Library which syncs with your iPhone, and others whose photos are stored only on your Mac (thus saving space on your iPhone). There is a catch: Photos isn't set up to work with two libraries at the same time, and to make it easy to move photos from one library to another. To move photos, you'd have to Export them from one library to the Finder, import them from the Finder to the second library, then delete them in the first library.