Android applications purchased in Amazon's app store (which may finally go live to consumers any time now) will use Amazon's special sauce of digital locker/digital rights management to help combat piracy. Amazon spells it out in its developer blog:
Customers who purchase an app will retain an entitlement to their app even if they decide to replace their current Android device and/or purchase new devices, as long as the new devices meet the installation requirements of the app. This provides insurance to customers that their purchased apps will be available for use on all supported devices, even if the customer has uninstalled or otherwise removed those apps in the past.
The digital locker service combined with a robust Digital Rights Management (DRM) solution not only make managing apps easier for customers, they also address one of the biggest concerns developers have: unauthorized copying and distribution. An authorized user can now install your app on any of their supported devices; however, if you chose to apply DRM on your app at submission time, your app will not run on unauthorized devices.
Any app that has Amazon DRM applied to it will require users to have installed and signed-in to the Amazon Appstore client to access the app. When an app is accessed by the user, it will verify with the Amazon Appstore device service as to whether the user has an entitlement to the app. If the user does not sign in or does not have an entitlement to that app, then the app will not be usable. However, any user can gain an entitlement by purchasing the app through Amazon.
The tl;dr version: You can use apps purchased from Amazon on multiple devices just fine; you'll just need Amazon's app store app installed for them to work. It's not nearly as scary as some would have you believe, and it's an extra cushion for developers. [Amazon Developer Blog]