This question has bothered me in watching Tangled as well; it's a bit of a plot hole. But it looks like the movie actually may imply an answer. Firstly, why would Gothel tell the child about birthdays at all? read more
This girl has been sheltered somewhat (confined to a tower for 18 years is a bit much when grounding someone, I mean, be thankful your parents didn't actually follow through on the grounding you until you were 30 per Mother Gothel's approach). read more
Therefore, Rapunzel knows about birthdays, and knows the ‘lights’ appear on her birthday, but Gothel does not initially know about the lanterns. It works. Gothel is always in the tower, not in town, for Rapunzel’s birthday, so the first few years she may not have seen or heard about the new lantern ceremony. read more