How to reference data between scenes

Static variables, once initialized, are shared across all instances of the class and it doesn't have to be attached to a gameobject in the scene.

Be careful if you set a static variable to an object that can be destroyed. The next time you try to reference it on a different scene will return a null reference.

It doesn't have to be initialized without a GameObject. But, you could use a static method to initialize it:

 public static int = Init();
 
 private static int Init()
 {
      return 0;
 }

But, for what you're doing I think the preferred method would be to create a GameManager.cs or GameData.cs script and have it as a singleton instance. And then to add the DontDestroyOnLoad() property so that it carries from one scene to another. This will have to be attached to an object in the scene. AND this will only handle the current session, you would have to add Save/Load data if you want to have the class be persistent on the next session.

 public class GameData : MonoBehaviour
 {
     public static GameData Instance;
 
     // Any data you want this script to have
     public GameObject playerPrefab;
     public string className;
     public int characterLevel;
 
     // initialize the Singleton instance
     private void Awake()
     {
         if(Instance == null)
         {
             Instance = this;
 
             // Carry over from one scene to the next
             DontDestroyOnLoad(Instance.gameObject);
         }
     }
 }