I just found something very interesting on the German site of Nvidia Shield on Facebook.
https://www.facebook.com/NVIDIASHIELDDeutschland/videos/286118175060421/?comment_id=286138091725096
To make it short, 2 people complain that Amazon Video is still not available on Shield even though it is working on Android TV already (Sony).
The employee answers that "they are continuously working on bringing the best and most famous Apps to Shield TV". Then someone else says (paraphrased) "doesn't Amazon prevent a release?"
Employee then answers " A little more patience :) ".
Now here is my crazy theory:
Since early this month Nvidia is claiming that the 3.2 update will "come towards the end of this month" or sometimes "end of this month".
Now how would they know this date so precisely so much days in advance? Either the certification process (including finding bugs) is already finished or very near completion or they are simply guessing which is a bad business move, because many people get pissed if a business doesn't hold their timelines.
After the whole getting Sony's Amazon Video APK working on other Android TV devices and then days later Amazon blocks the workaround story, it's pretty clear that Sony paid money to get an exclusive Android TV release.
When did these Sony Android TVs come out? End of Februar 2015 or early March depending on the market.
If Sony bought exclusive rights for 16 months (1 1/3 years), it would end exactly at the end of June 2016.
tl;dr:
Nvidia Shield Facebook Employee says "a little more patience" for Amazon Video App and Sony's exclusive rights could end End of June, coincidentally exactly when the 3.2 update is announced for.
It is very likely that Android as a whole will be getting it with Google announcing the changes to the way payouts are handled in the store. According to Amazon that issue was the sole hold up.