I equate a stirrup cup with fox hunting. Technically, it's defined as a"parting cup" given to guests, especially when they are leaving and have their feet in the stirrups, hence the name "stirrup cup." Sometimes a stirrup cup is used as a name for the the traditional drink, usually port or sherry served at a hunt meet.
Little is known about the provenance of the stirrup cup but it's generally believed that it's tied into fox hunting and is most likely English. Between 1670 and 1760 "dram cups" or "tot cups" were used to serve drinks to English riders. These were small two-inch silver handled bowls with feet and have been referred to as stirrup cups. You'll see that stirrup cups come in many forms and that they are most generally a small shape that can easily be held while riding and therefore don't need a foot like a regular glass.
(unknown source, Pinterest)
Here are two dram cups from the 1600's:
Stirrup cups come in many forms. These are Staffordshire:
These are more modern, what you often see today:
Stirrup cups are collectable and some use them today as jiggers. I had to include this, which I found, and I love it.