Yes, Apple has trademarked the App Store name, and yes, it is suing Amazon because the book seller is using that trademark. "Amazon has begun improperly using Apple's App Store mark in connection with Amazon's mobile software developer program,” Apple notes, while adding that Amazon will also use the name for its Android app store. "We've asked Amazon not to copy the App Store name because it will confuse and mislead customers," an Apple spokesperson told Bloomberg.
Amazon apparently didn't provide a substantive reply and I can't say I blame them. This has to be the most idiotic lawsuit between two major companies I've ever seen in the technology industry, and considering this industry is trying very hard to be as legally idiotic as possible, that's saying a lot.
'App' is short for 'application', and a store is where you sell stuff. The fact that a trademark has been granted for something as idiotic as putting these two things in a name further illustrates the utter incompetence of the US Patent and Trademark Office. I can only conclude that the people working at the USPTO have all undergone lobotomies.
Dare to prove me wrong.