>I thought I remember reading somewhere that there are terms for applications that have User Name and Password for log in and others that have only Password. Anybody knows these terms?

You would need to have uniqueness somewhere. So, assuming the later, if uniqueness is not enforced on the password, you could have two users having the same password and it wouldn't be possible to guarantee which users is in fact behind the keyboard.

Usually, username is unique. So, with that, that resolves the issue.