Autobiography; Coming of Age
Even though Asher Lev is a fictional character, he's still writing an account of his own life—last time we checked, that's called an autobiography. Add to that the fact that this novel is about Asher's transition from childhood to adolescence to young adulthood, and we've got ourselves an autobiographical coming of age novel on our hands.
The novel begins with Asher as a young child, discovering the heady freedom of being both a little kid and an artistic genius:
I remember those early years of my life, those first years of my efforts with pens and pencils and crayons. They were very happy years; laughter came easily to both me and to my mother. We played. We took long walks. (6.4)
The novel is structured around the chronology of Asher's existence. For the first twenty years of his life, we go where he goes, see what he sees, and essentially grow up with him. We follow him through his childhood into the tragedy of his uncle's death and his mother's depression, his struggles with his artistic gift, and his banishment from the community he loves to dearly.
The story ends with Asher coming into his own as an artist, which is a sign that he is becoming an adult—or, to put it more technically, "coming of age":
'The apprentice has become a master.' (362.3)
When Anna Schaeffer says this to Asher after his second art show, the implication is that he has outgrown his training with Jacob Kahn and is now ready to become a practicing artist in his own right. He's such an adult that he even gets forcefully emancipated from his family. Welcome to the real world, kid.