Kagi's shared threads don't offer an option to share publicly
Threads can be marked as public by assigning the Public tag to them. At that point, anyone with or without an account can view the thread - unless the author deletes it or it expires.
I have queued an update that will ask crawlers not to index these pages, which I think should help. Additionally I asked our team to see if we can issue a removal of these results.
Ultimately there is only so much we can do to prevent this, it relies on other crawlers being "polite" and honoring the request per their own documentation; unless we start issuing something like Turnstile to try and block bots/crawlers more aggressively. It will always be a cat/mouse game to some degree.
Another "middle of the road" option, is to add an additional level of "Public" that is "Public to everyone with a Kagi account". Subscription would not be required, you'd just have to be signed-in. This is probably the most effecitve way to stop shared threads from being indexed, however it will add friction, as unfortunately not everyone is using Kagi yet 🙂
Would people still find that useful? It adds a small amount of complexity to sharing, it adds another choice that users have to make when sharing a thread, and they'd have to understand why they'd want one option or the other.
Here is an exmaple from another service that I use:

(It's a collaborative note-taking tool, so the use-case is somewhat different...)