This app was mentioned in 9 comments, with an average of 1.00 upvote
I personally just use the website in Chrome, however, to test this for you, I installed the "Amazon for Tablets" and it booted up and let me sign in just fine. Just in case you're installing a wrong version, here's a link to what I installed:
https://play.google.com/store/apps/details?id=com.amazon.windowshop
Edit: Please disregard, I read it backwards, thought you were asking about the store app, which installs via play store, not side loading.
To whomever decided to down vote someone for a legitimate attempt at testing and helping... Well shame on you, at least I spoke up and attempted to help, even if misunderstanding the request.
I can only tell you that the method above has worked every time. Amazon Underground is a Marketplace, just like Google Play. I doubt Google Play offers you a direct competitor, so you probably installed some kind of other app, which sounds shady enough. Let's hope it was just: https://play.google.com/store/apps/details?id=com.amazon.windowshop which is an amazon shopping version (like the website), not a Marketplace for apps.
Is your device up to date?
You could always try installing the app using the APK file. Not guaranteed to work, and it poses a security risk.
Also curious, what does it say for compatibility if you look at the tablet app?
https://play.google.com/store/apps/details?id=com.amazon.windowshop
> This mobile application does not currently support digital downloads. Add this item to your Wish List and you can view it and purchase it from your computer.
On this app on tablet Samsung Galaxy Tab 4 7.0.
Which AP app?
The one I know works is the Android app for tablets. https://play.google.com/store/apps/details?id=com.amazon.windowshop
If you want it on a phone you'll have to sideload it. Even then it downloads a video player when you try to watch something.
List made using List My Apps
There is a separate Amazon shopping app for tablets.
https://play.google.com/store/apps/details?id=com.amazon.windowshop&hl=en_US&gl=US
They have a separate app made for tablets that runs fine on my Pixel C. Give it a shot.