Vlad If you use any email address that you control, it will always be linked to your identity, even if you use an alias service. If you register with a fake email, you will not have access to it, and someone else may take over your account (in theory if he controls that fake email address).
It's worth noting that the risk of losing access to an account already exists for anyone using a fake email. It's a personal decision to weigh the pros and cons of this method and out of Kagi's responsibility. As I suggested earlier, it might be better for Kagi to adopt a registration method like IVPN, which provides a user ID and optional email and password authentication (with benefits such as 2fa and account recovery). Also, just the possibility of selecting a username during registration instead of an email would be a benefit. If you collect less of users' personal data on your servers, you automatically become more privacy-friendly.
And please don't get me wrong: I know that resources are limited, and Kagi is young on the market. I don't expect any of the suggested features to be implemented within the following months. I happily use and will continue using Kagi without this feature. I know that the current registration method causes some concerns in the privacy community. I think an anonymous registration would reduce these concerns and would, in general, be beneficial to your service. Especially since you claim to offer anonymous searches in various places.