There's also
bar mitzvah 1: a Jewish boy who reaches his 13th birthday and attains the age of religious duty and responsibility
bat mitzvah 1: a ceremony and celebration for a Jewish girl usually on her 13th birthday when she takes on the religious duties and responsibilities of an adult; also : a girl for whom a bat mitzvah is held
Quinceañera a ceremony on a girl's fifteenth birthday to mark her passage to womanhood, to give thanks to God for his blessings, and to present a young woman to the community.
But like the, Roman Catholic celebrations, these are all specific to their culture and not used to broadly cover maturing of any kind. I mean to say they are not general terms that can be applied to just any other culture's version of a specific rite of passage. Your original question sounds to me like you're looking for something that is inclusive of all such recognition of maturity, but maybe not.