Mac asks for Photos update, but there is no update!

I wonder--is your Photos Library on the hard drive? How is it formatted?  To avoid damaging the Photos Library an external drive must be formatted in either APFS format or Mac OS Extended (Journaled) format. Additionally, the drive can not have had Time Machine on it since it was formatted. There have been so many problems with using incompatible drives that the newest macOSs won't even allow a Library on a non-Mac formatted drive to open, since there is a chance of damaging the Photos database. See this:

Move your Photos library to save space on your Mac - Apple Support

Frankincense246 wrote:

It certainly was formatted in the correct way as I used to be able to open it, and I don't use Time Machine. I'll check again and see if it will open on another computer

The previous versions of Photos allowed us to open a library on an unsuitable external drive, where the library could become damaged. Only the most recent versions are more protective and are refusing to open a library on an unsuitable external drive. It is not just the file system format or Time Machine backups. Any software installed on the drive can cause permission problems. And the drive needs a wired connection, not network connection on your local network.

An easy way to check if an external drive is supported is to try to set the "Ignore ownership on this volume" flag in the info of the volume. If this flag cannot be enabled, there is a problem with the volume, see:: Move your Photos library to save space on your Mac - Apple Support

How are you launching Photos to open the library? Are you clicking the Photos icon in the Dock or are clicking the Photos Library on your external drive?

After a system upgrade there may be an outdated item in the Dock or an older Alis on your Desktop linking to an older version of Photos, particularly, if the library is on an external drive containing backups. If you have used the Dock icon, CTRL-click it and try the command "Options > reveal in Finder" to check, which version of Photos the icon is linked to. If it is a an older version on a backup drive, remove the icon and add it again from your Applications folder.

Then you may want to try to open your Photos Library this way.

• Quit Photos, if it is running.
• Select your Photos Library in the Finder, then open the Info panel with the shortcut ⌘i. In the middle of the Info panel check the "Open with" info. It should be set to "Open with Photos.app".
• Then double-click the Photos Library to open it. It should automatically use the Photos.app version in your Applications folder.

If have downgraded from Sonoma or opened the library on Sonoma, you can no longer open it on a Mac with an older system version. Photos has no tool to downgrade your library. You would have to create a new library on your older Mac and import the photos again. If you are using iCloud Photos and your library has been synced to iCloud you can easily recreate an older version by syncing a new, empty Photos Library with your iCloud Photos Library.

The alert tells that some newer macOS version (maybe Sonoma) has previously opened and updated that Photos library. So to open it, you must update macOS (and Photos with it maybe from Ventura) to that newer version.