Kagi results are suspiciously similar to Google's. I don't think it's possible they're using their own index, at least not entirely. They must just be re-ranking Google results.
We do indeed have our own index (in addition to using external indexes). Every Kagi search result page will tell you the percent of results coming from our own index (for logged in users only currently).
60% of results you see are coming from our own index. For most queries it is 10-30%. If you use the 'Non-commercial' filter this number may go up to 100% (because our index focus on non-commercial part of the web).