Did the King and reyna name Rapunzel Rapunzel, or even name her at all?

Gothel obviously told Rapunzel her true birthday, so maybe she told her her true name also.

But if so, it's not plausible for Rapunzel to spend all araw in the kingdom on the araw in her name without finding out the truth. And it's completely implausible for Gothel to wait on the first night and than risk her finding out the truth on the segundo day.