Okay, so here’s the short version: You use your real name for your profile name – that is the one you can prove, or you can take a hike.
Yes, Google’s happy for you to use a pseudonym, nickname, maiden name, or inoffensive anatomical quirk as an “other name†(in the field provided) so long as you’re not using it as your profile name.
The implication is that that means “real names†only – or, more to the point, a continuation of the policy currently in place: A mix of requiring proof for some while making exceptions for some others, with those using their wallet-names apparently getting most of the collateral damage from the process.