I also might not want my previous searches to impact my later ones. (Just give good results again, Google Search...)
Can the search engine look at your IP address or other location information to give you more accurate local results? Can they build a profile of your previous searches, to determine what you're interested in? If you search for your spouse's name or your place of work, is that now data they can associate with you?
This is not an either/or question of "user data"/"not user data" or "consent"/"not consent", but shades of gray (how much consent are you giving, and to how much of your data and for which purposes).
My comment was just meant to illustrate that there's a spectrum to discuss rather than it being a black & white issue. I tried to do that by giving an example of another point on that spectrum.
I already made the point that a search engine can just search for the given string and not muck about with user-specific data… that’s what the user expects a lot of the time.