I’m not even remotely a legal expert and I don’t know what type of popup that is but I think the anti-competitive piece is “could Google use the same technique to push the user to switch to google search on Edge or not?”.
If this was an ad from a web page OP had opened or from the game and if clicking “Yes” only directed the user to a site with instructions on how to switch default search engine on Chrome, then yes, obnoxious but probably fair. Google could strike a deal with the game developers to push their search engine to Edge users or buy an ad. Someone writing a new browser or search engine will probably have considerably less money than Google but could reasonably do something similar to try and gain market share.
On the other hand, if that popup comes from Windows itself and especially if clicking “Yes” directly changes Chrome’s settings, then this is Microsoft using their ubiquitous (on desktops) OS to nudge more users to switch a competitor’s browser to their own search engine. Google, or even less a new competitor. would probably not have the same type of OS-level access to switch the settings of a different browser.