It's just a fanciful name for their content marketplace. The name making sense is less important than building the association between what the business does and the symbol that is the word.
Sometimes a fanciful name is better than a logical name because it decreases the risk of ambiguity among consumers (and so you don't end up in legal battles over very generic sounding names as can be seen with both Apple and Amazon using App Store). It also means their scope isn't limited by their name and they can move into new markets.
Apple, Amazon, Google, and many others all have company names that make no sense and I vastly prefer that to the more accurate names Consumer Electronics or Search Corporation.
Because they don't just want to sell apps. They want you to get all your digital content from the play store; music, video, ebooks and games/apps. Play seems a fairly good and obvious name for that to me.
The rebranding occurred when Google changed it from an Android app store into a one-stop-shop for apps but also books, movies, and other downloadable media.
I took it as a way of making Google's involvement more clear; it's not immediately obvious "Android Market" is Google-related, where "Google Play Store" is.
I've been thinking more about this, and it's strange that Microsoft also rebranded their Windows Phone Marketplace to Windows Phone Store. Makes me wonder if focus groups don't like the word "market" for some reason. It's not commonly used in Americanized English, so I wouldn't be surprised if many people are confused by what Market is, in terms of a phone application.
I am sure having just changed the name from "Android Market" to "Google Play", and "Google Docs" into "Google Drive," they wanted to avoid confusing the hell out of people by rebranding one more time for no reason.
The Android Market app store was the biggest part and served as the framework but Google Play wasn't a renaming of the Android Market but rather was created to merge Android Market, Google Music, Google Books, and everything else Google sold to consumers into one store. Android Market wouldn't make sense as a name because you don't need an Android device to buy anything, with the exception of Android Apps, on Google Play (though if you want one you can buy an Android device on Google Play). Google Play media purchases are all playable on your computer.
What? While I can certainly appreciate fun in the workplace, that is not the central tenet of most businesses, and even at the places where it might happen to be that way then you're still going to require some "serious" apps, such as for email, chat, remote desktop, server and asset administration, etc.
Sometimes a fanciful name is better than a logical name because it decreases the risk of ambiguity among consumers (and so you don't end up in legal battles over very generic sounding names as can be seen with both Apple and Amazon using App Store). It also means their scope isn't limited by their name and they can move into new markets.
Apple, Amazon, Google, and many others all have company names that make no sense and I vastly prefer that to the more accurate names Consumer Electronics or Search Corporation.