The problem is that the music playing object exists in the scene to begin with, so when the scene is loaded again, a new instance is loaded as well. Since you can't undo DontDestroyOnLoad() that means you have to either load it after the scene is loaded, provided there is no such object in existence already, or destroy the excess one after the scene was loaded.

I personally use this script here to manage all those objects, and tag them "DontDestroy". This script also manages applying the DontDestroyOnLoad() attribute to these objects so I don't have to put it in each individual script I want to keep around.

using System.Collections.Generic;
using UnityEngine;

public class HouseKeeping : MonoBehaviour
{
    private void Awake()
    {
       //make sure the game keeps running even if the application is out of focus so communication to the server/client continues
       Application.runInBackground = true;

       CleanUtilities();
    }

    /// <summary>
    /// Make sure we have only one instance of each utility
    /// </summary>
    private void CleanUtilities()
    {
       //Get all the utilities in current scene
       var utilities = GameObject.FindGameObjectsWithTag(transform.tag);

       //Will contain unique names of utilities
       var uniqueUtilities = new LinkedList<string>();

       if (utilities == null) return;

       for (int i = (utilities.Length - 1); i >= 0; i--)
       {
         //Search if we already have this kind of utility and destroy it
         if (uniqueUtilities.Contains(utilities[i].name))
          Destroy(utilities[i]);

         //If not add it to our unique list and apply DontDestroyOnLoad to it
         else
         {
          uniqueUtilities.AddLast(utilities[i].name);
          DontDestroyOnLoad(utilities[i]);
         }
       }
    }
}

You can use the singleton pattern. There is a lot of forum posts and questions in Unity Answers regarding this topic.

For example: http://answers.unity3d.com/questions/17916/singletons-with-coroutines.html