In this statement, the author is talking about the way in which the identity of "Japanese-Americans" is understood in the United States. A person who is Japanese-American is someone who has both Japanese and American parents. It can also refer to someone who has Japanese and American ancestors, or someone who is of Japanese background but born in the United States. In this case, the author is "third generation." What she implies in this line is that although her family has been in America for three generations, her Japanese identity continues to play a role, whether this is positive (due to her pride in her culture) or negative (because people do not see her as a full-fledged American).