does have a monopoly on search. Everyone knows it, even the Washington Post had an op-ed about it recently:
https://www.washingtonpost.com/opinions/yes-google-has-a-monopoly-whats-wrong-with-that/2020/10/20/d2c0c022-1311-11eb-ad6f-36c93e6e94fb_story.html
You're caught up on the U.S. legal definition of the word, but you've already lost that argument (and now this one as well) so I'm not going to relitigate it.