IOS – Getting iPhone 6s plus storage full alert message

Some research led to this article from AllThingsD about how an Apple rep had confirmed that the service does indeed stream the music, but it also saves to the device, thus taking up storage space. It starts playing before it finishes downloading, but it does actually download and save.

MacRumors and a video from InsanelyGreatMac are somewhat unclear about this, but MacWorld agrees that on an iOS device, it's downloading (scroll down to "Playing iTunes Match songs on your Mac and on iOS").

UPDATE as per comment: As of iOS 5, the device will delete cached data from apps when it's low on space. This caused some controversy at first because apps had no way to prevent cached data from being removed automatically. In the linked article, Marco Arment provides an example:

A common scenario: an Instapaper customer is stocking up an iPad for a long flight. She syncs a bunch of movies and podcasts, downloads some magazines, and buys a few new games, leaving very little free space. Right before boarding, she remembers to download the newest issue of The Economist. (I think highly of my customers.) This causes free space to fall below the threshold that triggers the cleaner, which — in the background, unbeknownst to her — deletes everything that was saved in Instapaper. Later in the flight, with no internet connectivity, she goes to launch Instapaper and finds it completely empty.

This was fixed in iOS 5.0.1 (Apple Developer documentation) so that developers can differentiate between cached data that is and isn't important.

So, the short version: Once free space gets below a certain level, the device will regain whatever space it can by removing cached data from installed apps. If it can't do that, or if you're completely out of space, it'll probably just tell you that you don't have enough space to do what you're trying to do.

You said you don't want to wipe the phone and reset it, but what about restoring it from a backup? During an iCloud restore, iOS re-downloads most of what it needs from Apple’s servers, so the process might reclaim your lost space, since iOS wouldn’t assign it to anything that isn't actually using it. After the restore, your phone would be returned to the same state you left it in.