I found this older bug:
https://kagifeedback.org/d/2814-safari-extension-redirects-all-google-requests
It appears the new version of the extension has the same problem. I am using the Safari Extension (just downloaded the latest, but the previous version was doing it as well). I have my Search Engine set to DuckDuckGo (as we used to have to select an unused search engine). Whenever I click on a link within a Google site, they use the "www.google.com?url=XXX" redirect before it opens the site - Kagi hijacks that and turns it into a search result on Kagi.com.
For example, if I am on Google Calendar, and I click on a link in an invite, it will do the above. I have to copy and paste the URL into the address bar to be able to get to the link. I tried "Denying" the Kagi extension on the Google Calendar and Gmail websites, but that didn't fix it.
If I am using a Google property like Google Calendar, and I click on a link, it should not do the Kagi redirection, it should load the URL in the browser. My assumption is that this is only a problem on Google properties since they use the "www.google.com?url=" redirect for all links. That uses the domain calendar.google.com. So one possible solution to this is just don't rewrite URLs to Kagi if the click came from a "*.google.com" domain.