Please help, someone is on my account.

[worldofneil]

I believe each d/l produces a randomly generated unique ID for the authenticator itself.

The download/apk is the same for everyone (possibly there's different versions for different resolutions, but given the basicness of the app, I doubt it). It's when you run and set it up that it connects to your SE account and creates the unique seed (either the server gives it a unique seed or it's some formula based on your IMEI your Android Device ID or a combination of them all). I know that if I manually extract the apk and data files with Titanium Backup and copy them across to another device it resets the data so it is at least checking the IMEI even if it it's not using it in the seed generation (That method would normally work for other applications).

I view the Kindle as a bastardized forked Android, so it's not really supported by SE.

It is still running a version of Android though, just with a different front end. If you can get hold of the APK then you should be able to manually install it. Either using an APK installer app on the device itself or using the Android Developer tools (specially ADB) to install it from a connected computer.

Edit: You won't even need to root the Kindle to install the APK if you can get it, assuming in the system settings/apps you can enable installing of 3rd party apps.