So the correct answer seems to be "we absolutely could link your searches to your identity, but we promise that we won't; however, you cannot verify that we are keeping that promise".
Which I'm afraid is not good enough. To be clear, I'm not saying that they are lying, and indeed I have no reason to suspect they are. But I'm looking for technological privacy, not contractual privacy.
I wonder if they'd even accept crypto, but their FAQ has nothing on their accepted payment and so it seems it needs an account even to just see that.
Don't rely on company promises if you can avoid it.
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.
For the record, I'm signed up for Kagi with a disposable email, though I didn't bother to anonymize my payment.
The short answer to your question is: yes, you can use it without linking any real-world identifiers to your search queries. That is explicitly stated in the linked privacy policy. With that said, I guess it's possible that they are violating their own privacy policy. That seems like a bad idea from a legal perspective, and I don't have any reason to suspect they're lying, so I'm not too worried about it.