Maybe it's because I was a kid when he was elected, but I never heard anyone question Barack's blackness. There was the birther thing later on but that was only white people who did that. And there were a few instances of people implying he was an arab Muslim ( like when Hilary's camp put out that photo of him in a turban) but that again was white people and that never really caught any traction.
Most black people didn't support him at first, but that was more because people assumed a black man had no chance of winning, not because they didn't think he was black.
Frankly, the difference between then and now is that if the opposition or a public figure said something like that back in the day, it would've just made them look dumb. People have less shame these days + political/cultural overton window is in a completely different place now.