From my understanding, all they require is a name, so someone could put down Daffy Duck as the name of the father and it would be accepted. Therefore, Lynette's surname could be absent from the birth certificate or she could have used a false name.
But the other point still stands which is that the MC may never have seen his birth certificate or have any interest in seeing it. Ask yourself, growing up, were you ever once been interested to see your birth certificate? I know I wasn't and I could easily believe that the MC feels the same way.
Neil has somewhat stonewalled him, whether intentionally or unintentionally, because of his waffling memory and you can tell from what the MC has said a couple of times now that he's all but given up trying to get any information out of him and he also seems disinterested in finding out anything on his own likely owing to the fact that he knows it upsets Neil when Lynette gets brought up.