No, Google Play, Amazon, and the Apple App Store allow installing the app on as many devices as you want, as long as all of those devices are using the same App Store account that purchased the app. The Apple App Store allows additional installs via family sharing to share your app purchases with your family.
The Android and Amazon apps will also work on as many devices as you want using the same login to the app. However, the Apple App Store app is unaware of purchases made on the other app stores and vice versa. A purchase on the Apple app store will not automatically enable your Android or Amazon apps.