The bt sport app is available on iOS, Android and Windows 10 - we've never had an application for Kindle / Fire tablets. As for our Android support matrix, we support devices that run Android 4.4 and above with the requirement that Google play services are installed in the device.
As far as I am aware there are no plans to support this additional platform.
You have access to BT Sport via your EE contract - in order to use this subscription, you are required to have your EE sim card in a device that you intend to use it. As far as I am aware, Amazon do not offer a Kindle tablet that faciliates 3G and or 4G.
Please refer to http://ee.co.uk/why-ee/bt-sport for more details.