I can log in to the bb_admin.php file (which I have actually renamed, as recommended) with my admin credentials. However, if I try to log in as a user at index.php with these same credentials, I get an error. The password was copy-and-pasted in both cases, so should be the same. My understanding was that an admin is also a user.

You are correct, you should be able to enter the same credentials forums-wide for the Admin.

However for the Softaculous setup, and any other miniBB-shared software hosting, perhaps you have to align what's stored in setup_options.php and what's in database. The only data which matters for the Admin login is stored in setup_options.php. Check it out for the most up-to-date real l/p for the admin.

For aligning the data, go to Admin panel — click Restore Admin data in database.