Amazon claims that "app store" is a generic term in its response to Apple's lawsuit over its use of the phrase, and that it therefore doesn't require any kind of license or authorization to use it.
Amazon's "Appstore" drew unwelcome attention from Apple last month in the form of a lawsuit seeking an injunction against its use of the term as well as unspecified damages. Apple claims the "Appstore" name infringes on its "App Store" trademark and is likely to "confuse and mislead customers," and further, that it's unlawfully using the "App Store" trademark to attract software developers to its platform.
Amazon, naturally, denies everything, including the claim that "App Store" - or "app store," as it prefers - can be trademarked in the first place. It acknowledged that it used the "app store" term to attract developers and that it did so without any kind of license, but "contends that no such license or authorization is required because 'app store' is a generic term."
The reply also notes the Oxford English Dictionary definition of "app" as "an application," and claims that Apple has in the past claimed it has "the largest application store in the world." It then goes on to quote Apple CEO Steve Jobs, who in October 2010 described Apple's App Store as "the easiest-to-use, largest app store in the world," presumably to illustrate that even Apple considers the term generic and interchangeable.
As Gamasutra notes, U.S. trademark law allows trademarks to become "genericized" and sometimes unprotectable, as has happened with trademarks like aspirin, heroin, thermos and zipper.
Amazon's full response to Apple's "App Store" lawsuit is available from the L.A. Times.