I’m running into this too and it’s annoying.
I think root cause is with the Kagi safari extension. The login token (personal private Kagi link) is saved by the extension, but the extension is only triggered when a search redirection occurs. So if you literally copy and paste a Kagi search url into safari private tab it will not log you in. You can also just go to Kagi.com and you won’t get the logged in page.
The safari extension needs to be fixed so that on Kagi.com domains it will attempt to log in.