I contacted Amazon about this back in early January and got this as part of a reply:
"I realize how convenient it would be for all our customers if the Cloud Player App is made compatible with Windows Phone, but please understand that "Windows" has exclusive rights to the hardware and software that would make it possible for Amazon.com to provide Amazon Cloud Player App for these devices.
Because of these restrictions, we are unable to offer Cloud Player App on Windows device."
This makes it sound like Amazon is claiming they need some reserved APIs to make Cloud Player work. Given other music streaming is available for WP, I find it hard to believe that. I do find the lack of a "we're working on it" in the response to be a bit disheartening though.