I think it's mostly due to the locations of these services. KuCoin, Huobi, Binance, outside of the United States, whereas Brave is. Also, they gotta make sure they're tightly following regulations. I mean, they have Brendan Eich himself as the founder and CEO. They definitely wouldn't want to ruin the reputation of such a big name not only in the crypto space, but also in the tech industry in general. Just my guesses though.
Isn't his reputation in the tech industry already tarnished due to his political beliefs and activities? Asking for KYC won't ruin someone's reputation in the cryptocurrency space, at least not in 2019, but it definitely leaves a sour taste in the mouth, when they could have relocated to a friendlier jurisdiction.