Interface for Shared Album Settings broken in Apple photos

Try the following in order as needed:

1 - as a test launch Photos with the Option key held down and create a new, test library.  Make it the System Photo Library, enable Shared Albums and test to see if the same problem persists. Does it? This tells us if the problem is limited to your current library or is more wide spread.

2 - open the original library with the Option+Command keys held down and repair the library.

3 - boot into Safe Mode according to How to use safe mode on your Mac and test the original library to see if the problem persists. Reboot normally and test again.

NOTE 1: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

NOTE 2: if you have a wireless keyboard with rechargeable batteries connect it with its charging cable before booting into Safe Mode. This makes it act as a wired keyboard as will insure a successful boot into Safe Mode.

Have you tried to click the blue (+) button over the name of the album?

On my Mac this button shows a dialog to add subscribers.