You can use a privacy-focused VPN like Mullvad to conceal your identity from the search engine. In most markets having a credit/debit card requires personal information so there's nothing you can do if they don't provide a privacy-friendly payment alternative.

Anyway, GP's question was simply:

"Can I use Kagi without linking any real-world identifiers (including credit card details) to my search queries in any way?"

And the correct answer is no.