No. what if you were Jamaican, or from Belese? That would lump all black people together as being "African' when that would be both insulting and untrue.
Any racial classification is going to be found offensive to somebody.
My point was just that a state might not simply reuse racial categories of a half-century ago when it issues it's short form birth certificates.
It's entirely possible that Obama's original long-form said "African." I was simply suggesting that the possibility that someone's original long form might read "Negro" or "colored" while what they'd get from the state today might say "Black" or even "African-American."