Though named "dragoon", the Japanese name (literally "PSICOM Aerial Dragoon") clearly refers to the historical mounted infantry units rather than the "Dragon Knights" of the Final Fantasy series.
"Dragoon" is the traditional name for a special soldier type in classical European warfare. Unlike in Final Fantasy, the original dragoons did not fight with lances, that being the job of "lancers"—a different soldier class. Dragoons were infantry that rode on horseback, but fought on foot. Originally they fought with a musket called the "dragoon," giving them the name. The dragoons' arsenal expanded to include guns, sabres, and axes. During the Napoleonic Wars they became medium cavalry and reached the height of their popularity.
After the Napoleonic Wars, to make better use of the number of men the European military had, the dragoons were trained to use lances and lancers became obsolete. Thanks to the advances of modern warfare, dragoons lost their niche and modern dragoon units exist only for ceremonial purposes.