Vlad I think what @TyroneHelmsman tries to express is that using a Mullvad like ID code or username for registration will in general reduce the amount of personal data required for registration and stored on your servers. And I generally agree with that position. If you stop asking new users for their email you don't need to store it either.
The thing with Mullvad is that unlike Kagi it is completely build for privacy enthusiast users who are usually quite tech-savvy. For a regular user, it is much more comfortable using an email for registration, as he doesn't need to remember different credentials for each website. So as much as I like the idea of having something like Mullvad's ID code for Kagi, I really don't think its practicable.
Something like IVPN would in general be ideal for Kagi, but still you would have to adjust it. Users should be clarified about the risks of not submitting an email during registration. And again, implementing this would require a lot of resources. Therefore, I think username registration should be provided by default and implementing this reasonable. Best case would be to offer a button to generate a username during registration like @TyroneHelmsman mentioned.