Unless you followed the instructions in https://brave.com/linux/#debian-9-ubuntu-1604-and-mint-18 (i.e. typing all of those commands in a terminal), then you’re not using our recommended packages. Instead you are using the Snap package, which is also maintained by us but which doesn’t yet work as well as the standard packages due to some technical reasons in the way that Snap packages work.
The reason why I asked this question (and why it would be useful if you’re able to capture a screenshot of that password prompt) is that it’s possible that it’s asking for your user password in order not to do something as root, but rather to unless the password keyring associated for your users. Normally that’s unlocked automatically at login time, but on my machine (Ubuntu 20.04 and 18.04 before that), I have had to unlock the keyring manually a few times in Brave, but also in other applications that store sensitive information in the keyring.