So Vivaldi as a Chromium-based browser also has personas/user accounts/whatever you want to call it, that shows your icon on the top right. It doesn't need to be an online account, and it allows you to choose a custom icon, even from disk (my Chromium install offers 24 icons but no "load image" picker). And... the custom icon can be an animated GIF!

So this is what the top right corner of my Vivaldi install looks like right now, but in real life it's animated: https://i.imgur.com/pqvhlsz.png