On the iPhone, use can set a name for comments and playlists in Music.app > For You tab> Account Profile > {Your name} >
You can set a Full name and a @Username there.
On the desktop, you can find a nickname at iTunes > menu Account > View My Account > Settings > Nickname [Edit], but it seems that is a different nickname than the above?!? This one is annotated as being used for reviews that you write.