Photo Library not working

First, make sure that Photos has "Full Disk Access" in System Preferences->Security & Privacy->Privacy:

Second, start Photos while holding down the Option key and select your library in the external drive.

It should work.

Photos library should be saved on case-insensitive MacOS Extended (HDD or maybe APFS although it is slower) or APFS (SSD) volume. In the past I have successfully saved and shared it at /Users/Shared but then there is a risk that other users might open it at the same time that might corrupt it so now I save it at the default ~/Pictures or on an external volume. External volumes should have "Ignore ownership" ON but this should be the default anyway.

Non-Mac formats like FAT might very well corrupt the library beyond repair although imported files might be laboriously salvaged from the library package.