Hi Vlad:
According to Kagi documents, that is not correct: https://help.kagi.com/kagi/faq/faq.html#why-does-kagi-search-require-an-email-address
It suggests a person could use "an anonymous email provider such as SimpleLogin" but that service provides a functional email address with a relay to a your real email address. So, Kagi instructions do not indicate that someone can sign up with a made-up email address (ficticious@domain.tld for example) which is what you seem to be saying: "something that looks like an email address."
If Kagi really does not require an email address, the sign up page should ask for the customer to create an account using USERNAME and password. Upon your system confirming the chosen username is not already registered in your system, you could then dump someone into their account page where they could OPTIONALLY enter an email address and name or whatever you might optionally ask for. Then at that point, if the user chooses to enter an email address, you do a proper verification routine before linking that email address to the user's USERNAME.
I think having a way for Kagi to contact me to let me know if a payment failed or about other account or service issues is a good idea. Also email is good for doing password resets. I don't think you should encourage people to sign up with a made-up email address or to use someone else's email address.
I'm just tring to be helpful here before you guys get too big to change things and have to deal with support tickets for email address issues. I do appreciate your privacy-focuses efforts in general!