This issue is happening to me now, since yesterday. When problematic update (2.x.x) rolled out it still worked the same in standard and private browsing. If there was a issue, then it was present in both standard and private browsing. Private mode did not behave differently in any way.
Yesterday I searched in private window on MacOS and Kagi asked me to login. Then i have opened private window on Safari on iPhone and when searching it asked me to login like it did on my Mac.
I have both extension installed on my iPhone so I tried to disabling first one then the other but issue persist regardless which extension is working. I did not change any settings, it just stopped working properly all of the sudden. Did not used private window from a while now so I'm unsure if it started after last update. I even tried to logout to try it with new token but no help.
Any advice would be appreciated.