It's hard for me to see how that would be correct. Background is where you are from, identity is what you identify as.
Background is what leads to your current cultural identity. The history of black people in America definitely shapes who they are today. Without that background they wouldn't be who they are.
Thus background is a critical and inextricable part of your cultural identity, and to try to separate it is meaningless.
It's very simple: if she says she's black, she's black, with all the 'background' and 'cultural history' and 'heritage' that go with that.
