Customize File Dialog Box and Add column to the grid

Short answer is - not possible.  

This dialog is not available for customization, except that one can only add some custom buttons to it.  e.g. below:

     public class FilesDialogExtender : IFilesDialogExtender

    {
        public void AddAction(PXGraph graph, string view)
        {
            if ((graph is PXGenericInqGrph) ||
                 (String.IsNullOrEmpty(graph.PrimaryView)) ||
                 (graph.PrimaryItemType == null))
            { return; }

            if (!graph.Actions.Contains("MyButton"))
            {
                PXButtonDelegate uploadFromDemo1 = delegate (PXAdapter adapter)
                {
                    var g = adapter.View.Graph;
                    g.Views[g.PrimaryView].Ask("MyButton", MessageButtons.OK);
                    return adapter.Get();
                };
                PXButtonAttribute buttonAttribute = null;
                {
                    buttonAttribute = PXEventSubscriberAttribute.CreateInstance<PXButtonAttribute>();
                    buttonAttribute.VisibleOnDataSource = false;
                    buttonAttribute.PopupVisible = false;
                }

                PXUIFieldAttribute uiFieldAttribute = null;
                {
                    uiFieldAttribute = PXEventSubscriberAttribute.CreateInstance<PXUIFieldAttribute>();
                    uiFieldAttribute.DisplayName = "My Button";
                    uiFieldAttribute.Visible = true;
                    uiFieldAttribute.MapEnableRights = PXCacheRights.Select;
                    uiFieldAttribute.MapViewRights = PXCacheRights.Select;
                }
                List<PXEventSubscriberAttribute> attribs = new List<PXEventSubscriberAttribute>();
                attribs.Add(buttonAttribute);
                attribs.Add(uiFieldAttribute);
                PXNamedAction.AddAction(graph, graph.PrimaryItemType, "MyButton", "My Button", uploadFromDemo1, attribs.ToArray());
            }
        }

        public IEnumerable<ToolbarButtonDescriptor> GetToolbarButtons(PXGraph graph, string view)
        {
            Type graphType = graph?.GetType();
            if ((graphType != typeof(PXGenericInqGrph)))
            {
                var btn = new ToolbarButtonDescriptor(null, "MyButton", "MyButton");
                btn.CommandArgument = view;

                return new ToolbarButtonDescriptor[] { btn };
            }
            return null;
        }
    }