In the post that explains why Microsoft chose to rename the music app to Groove (
Here's why Microsoft chose 'Groove' as the new name for Xbox Music | Windows Central ), they explain some confusion that users had understanding that they don't need an Xbox or Xbox account to use the music app. I can understand some people being confused over the naming convention, especially when Xbox wasn't used to describe any other media or streaming or entertainment options within Microsoft.
Groove isn't a bad name, it is a musical word, and it is more creative than a very bland Microsoft Music or something. Only deep Microsoft insiders will recognize that Groove is a recycled name, and they will not confuse a music streaming service with a former office productivity sharing service.
The more important thing here is that if Microsoft is going to stay in the music services arena, they have to get more competitive. When they introduced the Zune Music Pass, it was a screaming deal as there was nothing like it at that time. The ability to stream pretty much whatever you want, whenever you want, downloading tracks or albums for offline playback and free MP3s each month was pretty revolutionary. But music collection matching, cloud storage, and radio station creation and playback are now features that are done better on competing music services.
People gripe about Microsoft being too open to the other platforms, but I don't know if there is a reason for iPhone or Android users to pick Groove over Beats (Apple Music) or Google Music or even Spotify. And there should be. This is one service that Microsoft really should be getting out to all platforms, get as many users as possible, and make their offering better than the competition.