Amazon Fire tablets run Fire OS, which is a modified version of Android. It's likely that the Android app would work on a Fire tablet, but Kindoo would need to upload their app to the Amazon Appstore in addition to the Google Play Store (maybe they'd be willing to if there's demand, based on swigginton's experience working with them).
It's also possible to install Google Play Store on a Fire tablet and download almost any app, but the process of setting that up is a little hacky and probably not endorsed by Google or Amazon.