Referred to as a “pocket hanger,” this leather ID badge holder is hooked over the breast-pocket button on the Service uniform coat before buttoning the pocket. Its round design makes it suitable for wear with a slew of Department of Defense ID badges, from the Office of the Secretary of Defense to all the Unified Combatant Command ID badges (European Command, Africa Command, Southern and Northern Commands, etc.).
Google searches about this item reveal that pocket hangers came in a variety of shapes to accommodate different ID badges—but finding an image of an Officer or Enlisted Soldier actually wearing one of any shape is well-night impossible. For some reason, the pocket hanger seems to be most commonly associated with ID badges worn by personnel serving at HQ, U.S. European Command, but multiple sources indicate its use was not limited to that badge.
Note that using a pocket hanger for an ID badge rules out the possibility of wearing the maximum two badges on a single pocket because it’s not possible to space the two badges equally from left to right on the pocket.