It's simply just a regulatory requirement for them to let some of their users accept and send BAT tokens in a legal matter.
Do we know why, though? There are hundreds of exchanges which trade crypto without involving fiat, which don't need KYC. There are crypto gambling sites which don't need KYC. Why is Brave, which also doesn't touch fiat, different?
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.