You're confusing private and anonymous. Anonymous would be that no one knows you searched "cucumber." Private would be that someone could know that you did but it isn't shared with every single person for any reason. DuckDuckGo for instance is private in the sense that no one knows you searched what you searched, but Bing Ads knows the query to serve you a relevant ad. Signed-in to Google, Google would know your query and target an ad based on it, while adding it to a profile— in other words, it isn't private or anonymous. In a perfect world or using a search engine like searx, it's private in the sense that only the searx server knows what you searched and anonymous in the sense that Google wouldn't know who from that searx instance searched a query.