Is there a way to "own" an app so that if it becomes paid, you will already own it? 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.
Google doesn't allow a free app to switch to paid status. So if you download a free app and they later choose to charge for it, they'd have to do so via an in-app purchase, in which case there's nothing you can do about it beyond raising a stink to the developer and Android blogs. Something else developers regularly do is make a SECOND app available on the store, this time paid, and rename the original app to indicate its lesser status, then either stop updating it permanently, or turn it into a trial version. Either way, you'd be SOL in this scenario.

Also, I don't believe they have any sales that reduce app prices to zero on the Play Store (someone can gladly correct me if I'm wrong), unlike Amazon. There was one year they put a ton of apps on sale for $.10, but that's the lowest price I've ever seen a paid app discounted to. That said, if they did put it on sale for nothing, you would go through the purchase process and see the full price, along with a discount of the same price, which means it would still count as a purchase on your account. The same way they do regularly with movies and music. Those have all gone back up to full price, but I still own them.