They theoretically could do things to drive traffic. For example, if they surface the domain sooner than they might surface other domains, or change the way the URL is displayed.

Eg. User types "B" and whilst "bbc.co.uk" might be very popular, Brave could instead surface the less popular "binance.com" ahead of BBC, and potentially give it a different visual treatment, too.

I don't use Brave, so I'm not sure whether they do either of these things, though.