I don't think they anonymize the search, they just send it directly to Bing or Google. Anonymizing the search wouldn't earn them any money.
> [...] a search for “airpods” on Bing leads to a Search Encrypt search results page that has more ads than search results. A user would have to have the stamina to scroll through 10 text ads from Microsoft and then 5 image ads before coming to an organic search result