Sure! It's what they call Privacy Pass. It's implemented using an extension that's available on most browsers (including Tor). The extension uses a session to generate tokens This step is linked to the account. The tokens themselves are not linked to an account, but are used to access Kagi itself afterwards. When you search using Privacy Pass, the extension offers those tokens instead of a cookie, so you lose any user preferences you've previously set up. Those searches are not linked to an account.
More info: https://help.kagi.com/kagi/privacy/privacy-pass.html
You can also create an account with a throwaway email and pay with bitcoin, so at that point I think it's pretty good for anonymity
They seem to do quite a bit of indexing themselves. A good alternative that's from Europe would be Qwant, but they don't have a feature for searching the fediverse. It would be nice if it got added to fedi-search :)
Qwant and Ecosia are building a search index, but I don't think there is anything that can really compete with Google and Bing at the moment: https://betterweb.qwant.com/en/2024/11/08/ecosia-and-qwant-join-forces-to-develop-european-search-index/