At the moment the latest version of AnkiDroid 2.8.4 seems to be in the Amazon appstore: https://www.amazon.com/Nicolas-Raoul-AnkiDroid-Flashcards/dp/B004L1G7T2

Since I wrote the post you linked to the authors apparently decided to update the amazon listing. I would try this. Maybe they restarted updating regularly. I would use this until it doesn't work anymore.

https://www.amazon.com/Nicolas-Raoul-AnkiDroid-Flashcards/dp/B004L1G7T2 should work if you don't have the Play Store.

If you do have the Play Store, it should definitely work.

Speaking unofficially here (as always), I don't deal with releases, and the below issues started and finished before I was involved.

It's available on the Amazon store, but honestly, I'd recommend another non-Amazon Android tablet to run AnkiDroid unless you're technical and willing to get your hands dirty.

The app is available on the Amazon App Store, but it's 2.8.4, and the reviews imply some things aren't working (we're up to 2.9.6): https://www.amazon.com/Nicolas-Raoul-AnkiDroid-Flashcards/dp/B004L1G7T2

There have been problems with getting later versions onto the store, it might be worth bumping a thread on the mailing list to see if there's current interest in looking into this.

https://groups.google.com/forum/#!topic/anki-android/__XSLXaOo-k

https://github.com/ankidroid/Anki-Android/issues/5051

There were bugs (maybe intentional) on the tablet around sideloading (installing from sources outside the Amazon app store). If they're fixed, you can download the latest version from GitHub, and ideally get support/fix any issues you encounter. If you can't, you're stuck with an unsupported version of the app, with no way to update it.

I recently thought about buying a fire tablet, too. What I found was this:

Amazon Fire Tablets run a customized version of Android. So you need Ankidroid. There is a version of Ankidroid in the Amazon Appstore, https://www.amazon.com/Nicolas-Raoul-AnkiDroid-Flashcards/dp/B004L1G7T2

So it should run but the version on Amazon is quite old. It seems to get updates quite rarely and who knows if this will continue into the future. In 2015 one of the principal authors wrote "I did not send updates recently because Amazon's requirements are very time-consuming, for very few downloads (about 50 last time I checked, vs 460k on Google Play)". I doubt that Amazon Fires got so much more popular in the meantime. If the numbers continue to be like this I wouldn't bother to update the app in the Amazon app store.

On Android you could try to sideload applications or install the play store. But it seems to be difficult on Fire Tablets and who knows if this works in the future, see https://www.howtogeek.com/232726/how-to-install-the-google-play-store-on-your-amazon-fire-tablet/

On the other hand: Do you really need new versions? But with the upcoming Anki 2.1 some major changes might be coming to the sync-server ...