alt Hacker NewsIt kind of depends. See Kagi Privacy Pass ("Allows you to use Kagi Search with Privacy Pass, which cryptographically ensures that Kagi cannot tie that request to an account and allows for further privacy and anonymity."): https://help.kagi.com/kagi/privacy/privacy-pass.html
... which requires an addon to the browser, or for it to be built in specifically for that company.
That's not something companies like Matrix can use. If you're installing software already, why not skip the browser engine and install a full Matrix client instead?