It seems Google already /dev/null's this search term.

    Your search - site:claude.ai/share - did not match any documents.

Seems they forgot to noindex the /share directory or path.

That said, someone has to have shared these links publicly at some point, since there's nothing on the Claude.ai site linking to all these shared chats. If you shared the link to your chat on Reddit, Stack Overflow, Hacker News or some other public internet forum, Google is going to find it.

Edit: I've heard mixed things about this, but as far as I can tell, there's no evidence that Google uses pages viewed in Chrome/links shared in emails to/from a Gmail account as things they should index. So, it seems like these links had to have been shared publicly at some point.