OP, I know this thread is old but I just encountered the same problem.
It's a ridiculous workaround, but it does work. If your like me with hundreds of songs, it takes a very long time. You have to scroll through all of your purchased music just to select each song individually to download. The songs are are not grouped alphabetically. Unfortunately, that page is not mobile-friendly.
There you'll find all of your purchased music you can select to download onto your device. On the next screen, click on the Purchased bubble under Library at the top of the page. Then, click on the little headphone icon near the top of the page and select Music. Then, from those 3 dots in the upper right corner select Desktop Site. On your Android phone, you have to log in to Amazon Music through a web browser like Chrome. The Amazon support guy did give me a workaround, though. At least that's my take, and I'm sticking with it. They want everyone forced to use the crappy Amazon Music app so they can continually hound users to subscribe to more of the services. Amazon, which as we all know is one of the richest companies in the world, wants to be even bigger and greedier and not allow people to download their purchased music to listen to on their own devices. The guy told me it's basically happening because Google doesn't want Amazon to allow their app to download songs onto Android phones. When this happened to me last month, I called Amazon customer support.
