I use Hangouts pretty extensively for personal use. I also use Google Voice to have SMS messages sent there. I notice that in Brave, if I’m not actively on my Gmail tab (I don’t use any extensions, or the Hangouts page), then new messages don’t arrive. Once I click on the tab, new messages flood in.
The desired behaviour would be that these new messages arrive/pop up real-time, even when not actively on that tab. Then the tab would blink, etc, like it did in Chrome.