I feel like people are taking the "no labels, but I'm American" thing a little too literally.

She probably means that she just doesn't like the idea of these types of labels we use to put people into different groups. At the end of the day, we're all just people. If you were born in America, you're American by default. And since she can't trace her roots back to one single place, she prefers to just be called "American", because she's not JUST black/AA. She's also other stuff.
Also, taking issue with the fact that in the US, if you're any shade of brown and you can't easily fit into any other brown race, people basically just call you African-American.

Like, you can be dark and not be from Africa.