I’ve seen other apps do this well and most recently reminded of it by seeing Rider utilize it to render some extra status.
See:
When you have multiple instances of Unity open, it can become a game of remembering which is which and it would be nice to have a little icon with some extra indicator of what project is open if any. Unity already does add some indication to show when it’s doing work (importing, reloading, etc):
One idea is to have the default project icon connected in ProjectSettings to be shown on the dock’s icon too.