Microsoft is apparently aware of this behavior and is doing something about it: Neowin has spotted new Edge pop-ups that specifically try to dissuade users from downloading and installing Chrome, a change that I promise I didn't know about when I wrote about Microsoft's annoying promotion of Microsoft Edge literally yesterday.
If you use Bing to search for essentially any web browser, including Chrome, Firefox, Opera, Vivaldi, or Brave, a large ad for Edge appears both above the search results and in a giant box to the right of the search results.
And whenever you log into Google's services using Edge or any other non-Chrome browser for the first time, you'll get a "helpful" nudge about downloading and installing Chrome.