With ytmusic plugin, do you get 256kbps streams when playing a song? If so, then you are playing from your premium account. AFAIK, YouTube Premium includes YouTube Music Premium (but not vice versa).
To link the youtube2 plugin to your account, you need to setup Google API credentials from the Google Developer Console, but youtube2 is lacking in many library features due to Google API limitations, e.g. you can only access self-created playlists, not those of others which you’ve added to your library. I am thinking of rewriting the plugin to get rid of the Google API requirement and provide a more complete set of library features. But this will require rather extensive preliminary testing which I do not currently have the time for.