What claims does The Joy Luck Club make about Chinese, American, and Chinese-American identity?

In The Joy Luck Club, the theme of identity is presented as something that is constantly changing. In China, a person's identity is based on status, specifically, family status. A person is defined by the family they belong to, who they marry, a woman's place in the marital hierarchy, or whether or not a woman is married at all. In America, names aren't important, heritage is often swept inside.

The mothers and daughters we meet in the story each struggle with their identity.... are they American, Chinese, or both?