Can I set up phpMyAdmin so that a user has to login with a username and password and then only sees databases that are assigned to that username?
No, not directly from within your Cpanel because thats exactly what Cpanel is already doing for you. you cant sub-divide your own phpmyadmin down to user levels because this is just asking for problems and hacking, and is a big security no-no.
The best thing to do is to simply create your own scripts that can allow users to access/modify the parts of a database they need to as this can be controlled and audited as need be.
