Custom Editor: show asset folders in ObjectField

To provide an answer for this question. It is possible. Here is a working example editor window:

 using UnityEngine;
 using UnityEditor;
 
 class FolderTester: EditorWindow 
 {
   
     private DefaultAsset targetFolder = null;
 
     [MenuItem ("Window/Folder Selection Example")]
     public static void  ShowWindow () 
     {
         EditorWindow.GetWindow(typeof(FolderTester));
     }
     
     void OnGUI () 
     {
         targetFolder = (DefaultAsset)EditorGUILayout.ObjectField(
             "Select Folder", 
             targetFolder, 
             typeof(DefaultAsset), 
             false);
 
         if (targetFolder != null) {
             EditorGUILayout.HelpBox(
                 "Valid folder! Name: " + targetFolder.name, 
                 MessageType.Info, 
                 true);
         }
         else
         {
             EditorGUILayout.HelpBox(
                 "Not valid!", 
                 MessageType.Warning, 
                 true);
         }
         
     }
 }

You can limit the scope validating it after the folder has been chosen. Here is a complete PropertyDrawer I just made to keep folder references in the editor :

 using System.IO;
 using UnityEditor;
 using UnityEngine;
 
     [System.Serializable]
     public class FolderReference
     {
         public string guid;
     }
 
     [CustomPropertyDrawer(typeof(FolderReference))]
     public class FolderReferencePropertyDrawer : PropertyDrawer
     {
         private bool initialized;
         private SerializedProperty guid;
         private Object obj;
 
         private void Init(SerializedProperty property)
         {
             initialized = true;
             guid = property.FindPropertyRelative("guid");
             obj = AssetDatabase.LoadAssetAtPath<Object>(AssetDatabase.GUIDToAssetPath(guid.stringValue));
         }
         
         public override void OnGUI(Rect position, SerializedProperty property, GUIContent label)
         {
             if (!initialized) Init(property);
 
             GUIContent guiContent = EditorGUIUtility.ObjectContent(obj, typeof(DefaultAsset));
 
             Rect r = EditorGUI.PrefixLabel(position, label);
 
             Rect textFieldRect = r;
             textFieldRect.width -= 19f;
 
             GUIStyle textFieldStyle = new GUIStyle("TextField")
             {
                 imagePosition = obj ? ImagePosition.ImageLeft : ImagePosition.TextOnly
             };
 
             if (GUI.Button(textFieldRect, guiContent, textFieldStyle) && obj)
                 EditorGUIUtility.PingObject(obj);
 
             if (textFieldRect.Contains(Event.current.mousePosition))
             {
                 if (Event.current.type == EventType.DragUpdated)
                 {
                     Object reference = DragAndDrop.objectReferences[0];
                     string path = AssetDatabase.GetAssetPath(reference);
                     DragAndDrop.visualMode = Directory.Exists(path) ? DragAndDropVisualMode.Copy : DragAndDropVisualMode.Rejected;
                     Event.current.Use();
                 }
                 else if (Event.current.type == EventType.DragPerform)
                 {
                     Object reference = DragAndDrop.objectReferences[0];
                     string path = AssetDatabase.GetAssetPath(reference);
                     if (Directory.Exists(path))
                     {
                         obj = reference;
                         guid.stringValue = AssetDatabase.AssetPathToGUID(path);
                     }
                     Event.current.Use();
                 }
             }
 
             Rect objectFieldRect = r;
             objectFieldRect.x = textFieldRect.xMax + 1f;
             objectFieldRect.width = 19f;
 
             if (GUI.Button(objectFieldRect, "", GUI.skin.GetStyle("IN ObjectField")))
             {
                 string path = EditorUtility.OpenFolderPanel("Select a folder", "Assets", "");
                 if (path.Contains(Application.dataPath))
                 {
                     path = "Assets" + path.Substring(Application.dataPath.Length);
                     obj = AssetDatabase.LoadAssetAtPath(path, typeof(DefaultAsset));
                     guid.stringValue = AssetDatabase.AssetPathToGUID(AssetDatabase.GetAssetPath(obj));
                 }
                 else Debug.LogError("The path must be in the Assets folder");
             }
         }
     }