There is no out of the box solution. What you can you do is, for example: is create new form control MediaSelectionCustom. Copy code from MediaSelection form control (/CMSFormControls/Media/MediaSelectionControl.ascx) and play around with DialogConfig
Properties to configure pop-up dialog settings (hide tabs, set library etc, set starting path etc.)
/// <summary>
/// Setups selection control.
/// </summary>
private void SetupControl()
{
// Setup control
mediaSelector.DialogConfig.LibGlobalLibraryName =
"Name of the available media library. Set when only one media library should be available.";
mediaSelector.DialogConfig.HideWeb = true; // indicates if web tab should be hidden
mediaSelector.DialogConfig.LibStartingPath = "Starting path of media library folder tree";
...