In the Amazon Appstore, you can own any app, even if it is currently free. That way, if the developer later charges for the same app, you still own it. Similarly, if it is currently on sale for free, you can own it forever. This ownership survives uninstalls, device resets, and switching to new Android devices.
Is there any way to do this in the Google Play Store?
From what I can tell, in the Play Store, you don't actually own any app that you got for free or through a sale that reduces the price to zero. As soon as you uninstall it, reset your device, or switch to a new one, it seems like you lose it. Hopefully I'm mistaken.