Amazon's App Store and Google Play Store are distinct stores. Purchases do not carry over between them because they are competitors (in the same sense, you can't return an item to Walmart that you purchased at Target, even if they both sell it). You can install the Play Store on your Amazon devices, which allows you to use your Play purchases on the device (without jumping through too many hoops, like getting the apk and sideloading it).

This sounds like it's integrating with Amazon's app store directly so unless there's some way of sideloading Android apps using the underlying bridge to run them, Play Store purchases would not be available on Windows.

I figure the reason Amazon and Microsoft a ganging up in this way is precisely to address this issue: They want vendors to support a more portable mechanism for purchases that isn't tied to the Google silo.

Unless something has changed recently that I'm unaware of, that definitely isn't currently possibly on Android itself. I assume it won't be possible using the Windows-version of the Amazon AppStore either. But maybe this will cause the devs to start pushing to make that a thing.

Absolutely not -- the Amazon app store and the Google Play store are completely different entities. The fact that Microsoft partnered with Amazon instead of Google for app store integration is a strong indication of how serious they are about this whole thing.