I'm not black, but a friend of mine is an African American in the same way that I'm Asian American. Identity is tough for everyone, including children of immigrants. To compound the issue, names and phrases often used to talk about identity in our everyday lives are often vague. Back to my friend. He doesn't consider himself black, because of his direct link to his African heritage. In his mind, American blacks are people that are descendants of slaves and have a culture that is distinct from his parent's culture. Similarly, though I am an American and my parents are from the Philippines, I don't generally identify as Asian American, because to me, that is a culture that has developed primarily in California and to a lesser degree, in areas with a lot of Asian people from many cultures intermingling, often for generations. That does not describe my experience, nor does the experience of American black culture describe the experience of my friend. In the end, it's about how we choose to identify ourselves vs. how organizations attempt to identify groups of people, even when those groups might be to some degree imaginary.
My mom is of mexican decent and my father of german. When I check a box on a census form, I'm supposed to check "hispanic". I relate to my mexican heritage as much as I do my german, which is not that much. I'm neither. I'm an American. Growing up my father told me I was a germican. I always liked that... I'd check that box.