This is where you went wrong. You should have got her Barbie dolls that mainly look like her. First thing my wife and I decided with our daughter is that she will not have white dolls. Well, at least blonde ones. She was surrounded by white kids (nice kids), but it wouldn't be long until the "Identity Crisis" thing came in.
NOW, we did find a group of dolls that could not really be judged by ethnicity: Monster High dolls. These dolls were different, but different because they were children of famous monsters.
DO NOT, give a white doll to a Black (or colored kid). For the simple fact society has been trained to believe white is the ultimate human being to be. Check out the "Clark Doll Test" on Youtube and all of this will make sense.