What is the purpose of the story "Fish Cheeks" by Amy Tan?

The purpose of the story "Fish Cheeks" by Amy Tan is precisely as Tan's mother points out at the end of the story: that "the only shame is to have shame," speaking in terms of one's heritage. Due to a childish crush on the son of the American guests as well as insecurity, Tan becomes more and more ashamed as an evening meal goes on, constantly agonizing over how strange Chinese food and customs must seem to an American guest. Tan wants to leave the table and escape the situation when she is offered a fish cheek, which she sees as embarrassing. However, she fails to realize that all of the foods at the table are her favorites and how inconsequential the evening will be in the long run. The truly shameful behavior is not having a different custom but being ashamed of one's own custom.

The purpose of Amy Tan’s “Fish Cheeks” is to illustrate the conflicts that arise for 14-year-old Amy, as she navigates the process of cultural identification. Amy’s parents embrace the traditions and customs of their native China, while Amy longs to be like, and viewed as, a typical American teen.
Tan’s short narrative uses the description of one Christmas dinner to detail the scope and importance of these conflicts. Amy's burdensome task of straddling two disparate cultures is effectively brought to life through the various “horrors” Amy experiences throughout the meal. Her teenage crush, the minister’s son, Robert, and his family witness “the strange menu” Amy’s mom has created, as well as her relatives’ table manners as they, “licked the ends of their chopsticks and reached across the table, dipping them into the dozen or so plates of food.” Finally, Amy is mortified into silence when her father “belched loudly,” a Chines custom signifying satisfaction at the end of a meal.
After the meal, when Amy and her mother are by themselves, her mother tells Amy that her “only shame is to have shame.” Amy’s cultural heritage, in part, defines her identity, regardless of her desire to assimilate completely into American culture.

The purpose of Fish Cheeks by Amy Tan is to teach people not to be ashamed of their true heritage; it is who they are inside that makes them who they are.
This message is communicated in the short narrative about Amy, a fourteen-year-old Chinese girl who lives in America. She has a crush on the Minister's son Robert, and she is mortified when she discovers that his family is coming to her Chinese Christmas Eve dinner.
Unfortunately, Amy is a teenager ashamed of her Chinese family and its traditions: in this case, the Chinese food cooked on Christmas Eve. She desperately wants to fit in and shuns the food: "My father poked his chopsticks just below the fish eye and plucked out the soft meat. 'Amy, your favorite,' he said, offering me the tender fish cheek. I wanted to disappear."
She is also embarrassed by cultural traditions: "At the end of the meal, my father leaned back and belched loudly, thanking my mother for her fine cooking. 'It’s a polite Chinese custom to show you are satisfied,' explained my father to our astonished guests. Robert was looking at his plate with a reddened face. The minister managed to muster up a quiet burp. I was stunned into silence for the rest of the night."
While she is embarrassed by her heritage at the beginning of the narrative, she eventually realizes that her parents are trying to help her build her confidence in her identity and culture. Ultimately, she ends up feeling proud to be Chinese.