I am wondering if the operating system on the Kindle Fire would count as an Android-based OS.
Amazon FireOS bases on Android Open Source Platform (AOSP). As it is published under an open source license everybody can fork it and build an own OS. Note that there is one major difference between AOSP and "Google Android" (which is installed on nearly every Android phone you can buy): All Google services are missing. Starting with Google PlayStore and all Google apps this also affected invisible parts like Google Play Services (which are e.g. responsible for receiving push notifications for all installed apps).
According to Wikipedia relevant versions of FireOS are at the moment:
- Fire OS 5 is based on Android 5.0 "Lollipop"
- Fire OS 6 is based on Android 7.1.2 "Nougat"
- Fire OS 7 is based on Android 9.0 "Pie"
The complete mapping list of FireOS versions and Android version can be found here: https://en.wikipedia.org/wiki/Fire_OS#List_of_Fire_OS_versions