Other companies have exemption policies, or mechanisms to get approvals. Amazon does not. Your choices were work privately on your own project or leave (or risk being fired for violating the policy.)

Amazon was more aggressive (in my opinion) and much stricter than any other company I've ever worked at.

Edit: I researched a lot of the other tech companies. Here's Google's documentation, for example, which outlines an exception process: https://opensource.google.com/docs/iarc/

Amazon has an exemption policy.

Not to the game development policies.

Off the top of my head, there's company policies for employees to work on their own projects at Google, Facebook, and Github.

How often are they granted, and under what circumstances?

Hopefully a more recent Googler will answer you. In 2013 they were granted often, so long as they didn't compete with a Google product. I had to close my photo sharing site off to new users when I started at Google.