But he is a commissioned officer in an armed services organization with the exact same rank structure as the 1960's US Navy. Kirk is the equivalent of a O-6.
While the terminology might be different, Kirk is a "naval" officer.
You got it the wrong way around: the terminology is the same, everything else is different. Starfleet is not the U.S. Navy. It has some similarities, but you can't infer anything about Starfleet by refering to the U.S. Navy: they don't have the same origin, they're not active in the same century, they don't have the same objective, they don't use the same kind of equipment.