We don't have proof of Microsoft doing that with Edge specifically. But that's academic because we already know that Microsoft uses GDID to track website visits:
Microsoft’s records showed that at that exact same minute, a Windows device carrying GDID g:6755467234350028 had visited the ngrok signup page.
Whether it does it with Edge or the system HTTP libraries or whatever it really doesn't matter, what matters is that they do.
People consider the phone features too. For example Huawei has built a really solid following due to their phones' quality and that following didn't go away when Google started sabotaging their phones. Eventually banking apps started releasing versions for Huawei AppGallery (the Huawei Android app store).
So the Google monopoly can be bent if there's sufficient demand.