Presumably because, as they move out of beta, they want people to start paying to use their search engine. Thats the only way they can afford to be ad-free.
You could just allow a given IP address ? But that's just as trackable and has tons of downsides with using from other locations.
That's probably a worse business model, but it would be really interesting to hear what other monetization ideas Kagi considered or is considering.
Managing customer expectations on api key usage (esp. if that key is publicly visible e.g. URL parameter vs. HTTP header) is not worth it unless you have higher-priced products.
Also api keys would mean you might have to prevent re-selling, etc. Furthermore, they could still analyze api key usage to get the same historical data on you as if you logged in.
An individual account which isn't permitted to be shared is different.