Hello,
Just bought Duo this month. The experience with Duo has been up and down honestly.
Regarding the issue with Kindle app,
it has been reported on the web before and no clear answer has been provided.
It sounded like the app was working around September but, when I checked now it is not working. The book link does not work and can't buy Kindle Books through Kindle app on Duo.
The current Kindle app works well on iPhones and others, but not on Duo.
Updating, uninstalling, re-downloading, and restarting does not fix this issue.
Hope this kind of inconsistent performance stops soon on Duo or I may get tired of it soon.
Not sure why yours isn't working as designed, most had an issue at first opening of the Kindle App, but closing and re-opening fixed.