Apple's is "App Store", Amazon's is "Appstore" -- technically the first would not qualify for trademark protection, as it is a utilitarian description of the function (if they want to claim that they trademarked "app", they'll have quite a fight on their hands ... applet, perhaps, but no one uses it, just the slang for "application" that they want to claim as their own). B&N's "AS" is invisible, in any case.
I'm not sure if you can buy the app without registering a device. When I clicked, it wanted my cell number or email address and the first thing it did was load the Amazon Appstore App -- then, it let me get the game itself. You might could try with an email address and see if it will let you grab the game without actually installing the Amazon app.
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