Photos problem on external storage: “Unable to open library - Photos outdated?”

The only way to use the same Photos Library on a Mac with Photos 3.0 High Sierra and Photos 4.0 Mojave is keep the library in iCloud.

• If you are not yet using iCloud Photos (called iCloud Photos Library on older systems), sign up for enough cloud storage to hold your complete Photos library.
• Enable System Preferences > iCloud > Photos Options > iCloud Photos (library) on both your Macs.
• On your MacBook Pro with Photos 4.0:
  • Connect the drive with the library.
  • Launch Photos, if it is not running.
  • Enable iCloud Photos in Photos > Preferences > iCloud, and let the library upload to iCloud.
• On your MacBook Air with Photos 3.0:
  • Create a new, empty iPhotos ibrary. The new library will be a Photos 3.0 library. Name it "Photos Library MBA.photoslibrary". It needs a different name from the original Library, so you will not accidentally open it again with Photos 4.0.
  • Then make this empty library your System Photos Library in the Photos > Preferences > General by clicking "Use as System Photo Library"
  • Enable iCloud Photos Library in the Photos > Preferences iCloud.
  • Wait for the Photos Library to download from iCloud to the empty library on the MacBook Air.
  • You will now have a Photos 3.0 photo Library on the MBA in sync with iCloud and the Photos 4.0 library used by your MacBook Pro.
• The first syncing may take a very long time, depending on the size of your library. But the beauty of iCloud Photos is, that you can use the same Photos Library across devices with different system versions and keep them perfectly in sync.

If you do not want to pay for additional cloud storage, you can disable iCloud Photos once you have agan a Photos 3.0 library. Then cancel the iCloud storage subscription. It should suffice to pay for additional iiCloud storage only for a month.

I saved this as a new user tip with a few more details here:

Downgrading a Photos Library to a Previous Version of Photos with the Help of iCloud Photos

Are the two computers trying to access the same Photos library on the same external storage device? That wouldn't work, because Photos 4 will upgrade the library to its own format, and Photos 3 cannot use a library in the format.