If pressing the MUSIC DATABASE button doesn't do anything, there might be a problem with the button itself-- that is, with the switch inside the keyboard that's controlled by the button. I'm not technically-inclined in that direction, so I have no knowledge or experience at diagnosing or repairing keyboard problems, and am just making a guess about the switch.
If the warranty period has elapsed, and if you feel up to it, it might help to carefully open up the keyboard and use a can of compressed air to thoroughly blow out any dust, hair, and other detritus that may have gotten inside the casing. If you're lucky, that might take care of the problem. Otherwise, if the switch has gotten worn out then you might need to replace the switch mechanism.
Note that all the Music Database feature does is set up the keyboard with a particular style and style settings (such as the style variation, A or B, along with the tempo), a particular voice and voice settings, and any other miscellaneous settings (reverb type, etc.). In short, the Music Database entries are like built-in registrations that you can recall, without having to manually enter all the settings yourself and save them to one of the user registrations for later recall.
Consequently, in a worst case scenario where the Music Database feature is no longer working and can't be easily fixed, the simplest solution (aside from buying another keyboard) might be to ask another PSR-EW300 owner to select some MDB entry you're most interested in, then go into the Function menu and tell you what each function is set to, along with which style number, style settings, voice number, and voice settings were selected by that MDB entry. Then you could manually enter all those settings and save them to a registration so you'll be able to recall that registration in lieu of the MDB entry.