I was shocked to find out you can't do this. This seems in my opinion like baseline bare minimum functionality for a feature like tab islands. It's insane that a tab island just becomes a coloured dot and it renders the feature like 50% less useful than it could be if you're making me open and close each one just to look at what's inside.
I get why maybe someone would want it to be a dot and it could even be the default, but no way to just right click and rename is so painful.