I think the Amazon Fire is considered, by Amazon, to be in the Kindle family. Doesn't really matter. The Fire is Android-based. So it's possible that some Android apps will run on it. Even though the B2 app may not be offered in the Fire/Kindle/Silk store, it may still run.
Remember, the App is only a wrapper for the WebUI. The App finds the B2 without having to enter the IP address. Then it turns control over to the WebUI.