Hey there - I suspect this may be due to not having Widevine installed. We don’t install it by default
Try visiting netflix.com or hulu.com and see if a “puzzle piece” shows near the URL bar. Click this and you should be able to install Widevine. Once that’s done, go back to YouTube. It should work as expected
This should be the issue we have tracking it: https://github.com/brave/brave-browser/issues/1985