I use Kagi Privacy Pass to authenticate when incognito browsing, not for regular browsing (as I would like to benefit from the personalization I set up).
Currently, the extension is either activated for all windows or none.
I wished the extension had a separate toggle for normal/private windows, so I don't need to toggle it on and off manually.
There would be a second toggle or option in the extension, that controls the behaviour for normal and private browsing separately.
I just quickly searched and it should be possible to check if a tab is incognito or not.
https://developer.mozilla.org/en-US/docs/Mozilla/Add-ons/WebExtensions/API/tabs/Tab#incognito