What does your feature entail? What is it for? How will it affect existing workflows or user experience?
In MacOS, Stage Manager should show the icon of the Profile along with the Kagi icon (as it is in the Dock). This way, if the user has multiple profiles open it will be easier to switch to the correct one.
Related to this item: https://orionfeedback.org/d/4349-selected-icon-does-not-have-effect-in-recent-applications-ventura-stage-manager
What are the exact ways that you see a user using your proposed feature? Please go into as much detail as possible, and provide examples of how other browsers/apps implement this feature, if applicable. If your feature suggestion adds on to an existing feature, how would it work into it to extend its usefulness?
As shown above, when the user has multiple Profiles open, there is very little to differentiate them visually in Stage Manager. Implementing this feature would enable a user to click on the Profile they want without any further exploration.