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.
Yeah I was actually hoping for an answer that doesn't require blind trust.
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.
In that case, you really can't use any hosted search engine because they could be logging all your queries, the time you made them, and associating them with your IP and any other information they can get their hands on. Are you familiar with any search engine that is technically incapable of logging your IP???
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?"
There are pre-paid credit cards you can buy in Safeway and such for a small premium. I'm not sure how to use them for paying online though (i.e. how to provide name/address/phone that match).
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.