"Two Kinds" is an appropriate title for this story, because it illustrates the clashing of identities between June and her mother, Suyuan.
Suyuan would like her daughter to be an American child prodigy. It really doesn't matter what kind of prodigy, though. Singing, dancing, playing music, whatever. Suyuan simply wants to live through her daughter and wants June to be something akin to the next Shirley Temple. In order to prod June in the direction of famous child star, Suyuan is constantly testing and pushing June with extraordinary tests. June constantly falls short of her mother's expectations, so June resolves to not try so hard to please her mother by being something that she is not. That is June's first step in self identity and self realization. She is choosing her own path, not her mother's path.
As the story continues, Suyuan decides to hire a piano teacher for June. After a bit of training, it's clear that June is no piano prodigy. Suyuan won't hear any of it, though, and invites a group of ladies over to hear June play. They are all shocked at the lack of quality to the piece, but the deaf teacher thinks it's great. At this point, June thinks that her mother will drop the piano practicing push. Suyuan does not drop it, and a few days later reminds June that it is time to practice. June tells her mother that she will never be the daughter that Suyuan expects. Suyuan responds by telling June that there are only two kinds of daughters. Those that obey their mothers, and those that follow their own minds. In a climactic moment, June seizes control of her identity and shouts out that she wishes Suyuan were dead.
That moment was a breaking point in their relationship. It was the moment that Suyuan lost her influence over her daughter, and it was the moment that June took full ownership of her identity. Before that moment, it was a power struggle. The theme of self identity is present throughout the story, but it is a theme that grows in its intensity until its final release.