In the past, there has been confusion about whether or not someone needs to register another person in their Pal Pad with their correct, exact, in-game trainer name, and there has never been any definite information one way or another on the matter.
The answer is that no, you do not. In order to battle or trade with someone over wi-fi, all that is required is that you both have the correct friend codes entered into your respective pal-pad. I hope this helps anyone that was confused like I was!