I'm actually super surprised Google does this kind of deals. I've thought they don't do this, becuase it would help competition to get started. Once you get traction, then it might be possible to drop the Google access.
This is what Kagi says in the FAQ.
"Our searching includes anonymized requests to traditional search indexes like Google and Bing as well as vertical sources like Wikipedia and DeepL or other APIs. We also have our own non-commercial index (Teclis), news index (TinyGem), and an AI for instant answers." [1]
I'm guessing Google does not this kind of deals unless very rare exceptions like Ecosia. You have some companies parsing Google results, like SerpApi, but it is a bit shady.
This is what Kagi says in the FAQ.
"Our searching includes anonymized requests to traditional search indexes like Google and Bing as well as vertical sources like Wikipedia and DeepL or other APIs. We also have our own non-commercial index (Teclis), news index (TinyGem), and an AI for instant answers." [1]
[1] https://kagi.com/faq