Mixed people often feel like they're being pulled from both sides; I have a mixed friend (her mom is German American, dad is Nigerian) who describes it as constantly walking in and out of two different worlds. But ultimately black, Asians, etc didn't create the one drop rule, or establish a society where a non-Hispanic/Indian/etc brown person is universally considered black in this contry. Black people didn't claim Tiger Woods first, the media labeled him black. Fuzzy Zoeller's racist comments aimed at Woods didn't mince words in terms of what color he thought he was.
Personally, I believe mixed people should be allowed to choose how they "label" themselves. But the minute they walk into a room they're automatically "black" in the eyes of most people. Equating this as an issue of "both sides" being wrong is ridiculous.