Anyone is welcome to sign up using their real name rather than an alias. However, common names will have the same problem: John Smith, John.Smith, JohnSmith, John-Smith, John_Smith and so on. Since every country has a "most common name" it will happen a lot. Also, many people prefer to post on forums using an alias rather than their real name.
If we hide the user name (I assume you mean that all posting would be done anonymously), we would run into the problem where people would feel more free to flame/insult/otherwise break the rules or that we simply wouldn't know who to respond to.
We do not allow members (or moderators, even) to change their own names because that is rife for abuse - you could keep changing your name and that would make quoting impossible and would also make the continuing sense of community impossible.
As with any place that allows you to pick your own name, there is always a risk that someone else will have the name already.