No proof required. For that matter, depending where you'd been born and when, you might have found it extremely difficult to lay hands on a copy of your birth certificate.
It would be possible to be prosecuted later on if you were found to be giving false information, some examples:
https://paperspast.natlib.govt.nz/newspapers/PWT19240130.2.58https://paperspast.natlib.govt.nz/newspapers/AS19230727.2.45 https://paperspast.natlib.govt.nz/newspapers/CHP19320831.2.112However just using a different name than on your birth certificate wasn't illegal, it had to be for the purposes of deception (as mentioned in the second link above) - even though the person in question had changed his name to hide the fact he'd been in prison, he had the right to do so and hadn't committed a real offence according to the judge.
For example, someone who was born illegitimately as John Brown but had gone most of his life by John Smith because his mother married a Smith when he was young wouldn't have been seen as committing an offence. There's a question and answer about that very fact here in 1929 from a "Moera":
https://paperspast.natlib.govt.nz/newspapers/NZTR19291017.2.41The usual reason someone would end up in court over giving false information at marriage would be cases of bigamy.
