Thanks for telling us that you've check the formatting of your external drive. You haven't told us what else you've checked. Do you use iCloud.com to sync your pictures?
Have you looked at the info window for the drive to verify that "Ignore Ownership" is checked? Have you had Time Machine on this drive since it was last formatted?
What else have you tried? For instance, are you having a problem only with this one library, or do other libraries fail to open? How about these things?
- Restart the computer (of course)
- Rebuild your Photos Library by closing Photos and re-launching by option-command-clicking the app icon.
- Make a new user and see if the same problem recurs with it
- Use another library if you have one, or make a small test Library with a few pictures in it and see if the same thing happens.
- Start in Safe Mode (different for different computers) This bypasses certain potentially disruptive processes.
Start up your Mac in safe mode - Apple Support
One of these steps may fix the problem, but their failure to work is also diagnostic.
Let us know if any of this works, of if not...
