That's nonsense. It's just Flash. Linux has no more trouble with it than any other platform. That includes all of the usual complaints about performance and crashing.
Nope, here is the service:
And here is the message I got when I tried to access it in the latest Chrome on Kubuntu:
If you're using the Chrome browser with Linux, you must disable PPAPI to continue using Amazon Instant Video. You can also use a different Web browser, like Firefox. hundreds of other compatible devices.