I was wondering, if my documents have another purpose aside from identification?
In fact, they have it for us. Fro example, we may use your U.S passport with double end: Proof of citizenship and individual identity.
Do I need to provide my U.S. Passport even though I lived over seas?
You can show any document that proves you lived abroad for a long time and you don't have a social security number. You could use your previous or current passport, school or employment records, etc.