Are there any figures of note to whom we ascribe significance to their birthday? Moshe rabbeinu is one, but do we know the birthday of literally any other tzadik?
Why is there no mesorah for acknowledging birthdays before the last lubavitcher rebbe? My rebbeim often said that the only person’s birthday mentioned in the Torah is Pharaoh…hamayvin yavin.
We acknowledge yahrtzeits. Why? Because that is the day that the tzadik had finished his accomplishment and was “niftar”, which means dismissed…not death as we usually think of it. We are sad when we lose a tzadik because we need them, but after time we use that very day as a celebration of their achievements…. Avrohom avinu was “bah bayamim”, he had come with his days, meaning he brought all of his days – every one of them was equally valuable, used to their fullest potential.
We don’t find a significance in general to birthdays. I’d suggest that moshe’s birthday is noted by chazal because it was a venoihapachu from haman’s goral; a way of Hashem toying as it were with haman harasha.