In the Microsoft world, "Applications" were known as, "Programs". Apple has always called its, "programs", Applications.
Apple truncated the word, applications, to, App. Then called their new, smaller (file size) applications for iPhone, Apps. Then filed for trademark status of that name, in The App Store.
Microsoft would, theoretically, call their program/s, Prog/s. And hence, The Prog Store. Doesn't sound quite that nice rolling off the tongue, but it would make sense.
Amazon couldn't come up with their own nifty name, so they just "borrowed" Apple's name for their store. Maybe Amazon should have called their store, The DroidApp Store, Drapp Store, Roid Store, The Amazon Store for Little Programs to Run on Android Platform Phones Store.
Just my early morning thoughts. No legal eagle here.
First of all, this little myth you are repeating has been circulated for a while now and its just not true. There are various examples of BOTH companies using BOTH terms throughout the past 30 years.
Second, neither company invented these words. They have been around hundreds of years and are just now being used in terms of computers.
Third, you can't trademark an abbreviation.
Get real