@Josh I got the following response from Amazon. I think your hunch is right, they do use restricted profiles.
But this still doesn’t help me with the problem, since the user can’t restore IAP purchases from within the profile, I can’t unlock the content for them once they are in the profile. Any thoughts/suggestions would be much appreciated.
Thank you for writing in. This will happen on latest 3rd Gen devices as Kindle FreeTime is built on top of multiple android users, which was a feature introduced in Android 4.2. This means that when the customer starts Kindle FreeTime, they actually switch to a separate android user. Data is not shared because Android allocates separate data directories for each user; ex “/data/user/0” for the parent, “/data/user/10” for the first child, “/data/user/11” for the second child, etc. I believe that the “sdcard” is also emulated per user in /storage/emulated/<android user_id>.