Cfrtjryk

I understand that the expression 'coming of age' may refer to both the legal and natural definitions of a young person that becomes an adult. The age specifically may be different according to the different contexts, but what are other common expressions both in legal or everyday language that are used to refer to this specific transition in life?

In the class of “other common expressions both in legal or everyday language that are used to refer to this specific transition in life” [and some other transitions besides coming of age], one also finds:

• rite of passage, “(anthropology) a ceremony or series of ceremonies, often very ritualized, to celebrate a transition in a person’s life. Baptisms, bar mitzvahs, weddings and funerals are among the best known examples.”

• age of consent, “(law) The age at which a person is legally considered to be mature enough to engage willingly in sexual intercourse.”

• age of reason, “(Roman Catholicism) Seven years of age, at which age a person is morally liable for the sins that he or she commits.”

• age of discretion, “(Roman Catholicism) Alternative term for age of reason”

• age of judgment, “The age at which a child is presumed to be able to judge the difference between right and wrong”

• age of majority, “(law) The age at which the rights and privileges of an adult are legally granted”

• legal age, “Alternative term for age of majority” [This term was mentioned previously by Neil]

Also, when you're talking about someone's coming of age experiences, you could also describe them as 'formative experiences'. When I was studying "The Motorcycle Diaries", this was one of the ways I was taught that I could describe the experiences that led to Che Guevara becoming more mature

Legally, one reaches maturity, the age of majority, age of legal consent, or emancipation.

The time when the law allows persons to do acts which, for want of years, they were prohibited from doing before.

Figuratively, one blossoms, matures.

Other synonyms for coming of age: to reach maturity (the end of adolescence physically).

To reach one’s majority, to be of legal age.

It can have different meanings for different contexts. To Kill a Mocking Bird is described as a coming-of-age story (bildungsroman), where Scout is only seven and is just starting school, hardly at the age of majority. Here, the phrase describes a significant step in maturing rather than the reaching of maturity.

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.

vision quest is a rite of passage in some Native American cultures. The ceremony of the vision quest is one of the most universal and ancient means to find spiritual guidance and purpose. In practicing cultures, vision quest is said to provide deep understanding of one's life purpose.

Rumspringa Amish coming-of-age ritual, the rumspringa--the period of "running around" that begins for their youth at age sixteen. During this time, Amish youth are allowed to live outside the bounds of their faith, experimenting with alcohol, premarital sex, revealing clothes, telephones, drugs, and wild parties. By allowing such broad freedoms, their parents hope they will learn enough to help them make the most important decision of their lives--whether to be baptized as Christians, join the church, and forever give up worldly ways, or to remain in the world.

But like the, Roman Catholic celebrations, these are all specific to their culture and not used to broadly cover any celebration related to going through puberty that might exist. 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. So anyway...