Two iPhones, one computer, one music collection
The iPhones may use separate Apple IDs and share the same music library from itunes, (as you said, all of it is rights free / downloaded from Apple). The Apple ID only comes into play when syncing contacts, calendars, and alternate Apple Services. You shouldn't have any problems with Music syncing between iTunes and your new iPhone. Furthermore, you may download Google Play Music on your phone, and use the already synced content. The choice is yours. 

All the best!