Being American is a nationality. Being Asian is ethnicity or your ethnic group. You can belong to the Asian ethnic group but be born in America to parents that were also born in America which would make you American. They don't cancel each other out.
"The state officially categorizes its population into six groups: white, African American, Native American/Alaskan Native, Pacific Islander, Asian, and Native Hawaiian. From those groups, Americans identity with ethnic groups that are even more specific. More Americans specify as German than any other ethnicity.