While not well articulated, the OP takes a position with which I agree (I apologize intensely for being "white").
In this society, the word "Black" refers to the descendants of American slaves. It is applied to, frankly, anyone who has even a tinge of "negricity," no matter how slight, even if the person in question could "pass" under the right, or all, circumstances. Hence, we have the fairly recent phenomenon of "Black" women who look nearly-white, and who take every imaginable cosmetic measure to close the gap fully between appearing "Black" and appearing "white" (e.g., Halle Berry, Meghan Markle, Rihanna, and now Senator Harris), while trumpeting and reveling in the cultural "fact" of being a Person of Color, or "Black." (Can you imagine any of them with a "'Fro"?)
But to the point, Barack Obama was not, culturally speaking, "Black." He is not descended from American slaves, never lived in a ghetto, only learned to speak in a Black patois (when it was convenient to do so) by copying it from others, and so on. It is for these reasons that Joe Biden famously observed that he looked "clean" and spoke like a normal [white] person. Why would one expect otherwise? Again, as the term is currently defined in our society, Barry Soetoro is not "Black" - any more than lily-white African American golfer, Gary Player is "Black."
Similarly, Kammy Harris is not "Black." She is not descended from American slaves (indeed, it appears she is descended from slave OWNERS), never lived in a ghetto, doesn't naturally speak with a Black patois, and was brought up by two PhD's. And while she would never admit it, I imagine she was ostracized by the bigoted kids of her childhood both Black and white, for being so different from both. Kids are cruel bastards, like it or not.
I wonder if - I don't suppose I will live long enough to see it - when we finally do get a Black President who IS descended from American slaves and has dark complexion, any Black people will finally acknowledge that Obama and Harris were "not really Black."