You can't prevent users from trying to register with the same name. Name isn't unique. Username and Email Address are however unique. If you set username to not display on registration and you've set the Username Fallback to Name then you'll run into issues as it won't force the username unique. Username Fallback should be Email Address or a custom field like a query or code field where you can add custom generation of username as at this time it won't add anything unique to the username when a duplicate is found.