There isn't under Natural Moral Law, although it is mostly forced if you use Common Law, which is closer to Natural Law than Positive Law. The reason is because most other people will want you to have a name in order to sign a contract. But you can just sign, if the other party accepts, with a description on how to find you, for example "the blond dude living near the Colorado river".

Under Common Law, yes, you have a method to copyright a name, or a trust, or a company, or whatever you want. This is necessary to distinguish you from other people that don't know you well, and therefore enter into contract properly with some guaranties. But a note must be said right at this point: Common Law is not Natural Law. People decide if they want to adhere to the Common Law jurisdiction or not, but they cannot choose if they adhere to Natural Law (you are always under it). That said, Common Law is waaaay better than Postive Law (State Law).