Actually I just tried using the session token on Chrome as well. After performing a search with the session token (as opposed to logging in) Kagi did not show up in the list of available search engines.
However I did finally figure out how to get the session token url to be used as the default search engine on Chromium based Android browsers.
What I did was paste the session token url into the address bar.
Then I opened a new tab, and went to kagi.com I was already logged in because of the session token.
Then I performed a search in the search bar on the homepage of kagi.com
I then checked search engines in the settings and Kagi appeared as an option, so I picked it as my default search engine.
I cleared all my browsing history and cookies and closed the web browser.
Then I opened the web browser and typed a search in the address bar.
SUCCESS! The browser is using the session token url as the search engine.
I tested this on both Brave and Chrome and it worked. So maybe this might help anyone who always clears history in their Android Chromium based browser.