Understood. If you modify the login form to show only the username, then you can set the password to be the same internally.
Setting the password to be the same as the username would be done when registering the user - you haven't explained how you register users. One easy way to achieve that would be to have an external registration page tied to a non-Moodle database which is then used to authenticate.
May I ask why you want to do something like this? Is it perhaps for very young children? On the face of it it sounds, shall we say, unsafe. Perhaps there are other solutions to your challenge?