Well, Browsers don't have much control over a PC right now. There is still an option like "Continue running background apps when Brave is closed" and that's the only way an extension can run like that.

I have only downloaded an extension that ran in the background like that, and then I uninstalled it because I didn't care about it, I didn't even understand why it had that feature to be honest. But it is rare to see extensions doing that.

Uninstalling the extension should be enough. there shouldn't be any trace of it and extensions can't really change much in a browser or computer to really affect your computer like if it was a native executable file with access to registry and such. Think about this, Browsers don't even have admin rights (unless you elevate it yourself) so it is almost impossible it could do something to your computer.