Looking at their privacy policy they state the following:
> We may store web requests made by user browser temporarily, with strict retention periods, for debugging purposes, and in a manner that they are not linked to an account.
For 2 years they've said "Kagi has zero marketing budget and spends zero on marketing", that's a lie and they still haven't changed it. They've sponsored over 200 youtube videos https://filmot.com/search/%22brought%20to%20you%20in%20assoc...
I asked because the search you've linked to only seems to list relatively niche chess videos on one creator's account. And the videos themselves are also ad-supported. It's (at least) not impossible that the sponsorship association is not budgeted in the way that (I think) you infer.
https://kagifeedback.org/d/3822-kagi-is-sponsoring-youtube-v...
"I watched Daniel King for 10 years and have previously donated to him, now I am happy that we are in position to sponsor his work which I believe is extraordinary.", this also definitely sounds like monetary compensation to me.