It depends what music player you're using. There is a default Android library which is a bit limited (in particular, it doesn't know about Album Artist) but some players ignore it and create their own database. Even the stock Google Play Music bypasses the Android library!
Using the Handheld Sync facility, you can specify that MC copies files over the default music folders in Android and also save playlists, and this happens automatically and the playlists will be visible on your Android music player (although I've found one music player that doesn't find exported playlists). I'm not sure what would happen if you use a drag 'n drop method, whether just the playlist is copied over, i.e. a list of files, or whether the files themselves are copied over and appear as individual files, or if the files are copied over AND a playlist of them is created.
