I see the Google Play Music app is available for iPad. Even see some user complaint in the store about it not being optimized for the iPad. When I search for it in the App Store on my iPad it doesn't show up. Does it exist?
Google Play Music is an iPhone app. You can still download and use it on the iPad, but it will work exactly like it does on the iPhone, just blown up to the bigger screen (or small in the center of the screen).
To see and download it in the App Store on the iPad make sure you've changed "iPad Only" to "iPhone Only" at the top left, after you do the search.
There are some third party apps for the iPad that will 'probably' let you listen to Google's Music service. No idea how well the work.
The ability to run iPhone apps on the iPad is from when the iPad was new. It wasn't ideal, but it was far better than being limited to the much smaller number of iPad only apps we began with.
Apple has never seen a reason to take this away, thankfully. Sometimes it is the only way to get apps from developers that choose not to create universal or iPad specific versions.