Oh it counts alright. They wear a patch that says "Nation's First" because of it. I suppose you could have a category for the unit that has retained it's designation the longest. Designations change from time to time due to periodic reorganizations of the entire military. For example, 182 is a "regimental" designator from when the Army last decided to use regiments (WWII I think). As I said the unit was originally the "North Regiment" of Boston from the King Philips War days. There could be an argument for the oldest active duty unit, which the 182nd is not.
A quick internet seach found this on army.mil
The oldest unit in the active Army is the 1st Battalion, 5th Field Artillery, which perpetuates the Alexander Hamilton Battery of the Revolutionary War. The oldest regiment in the active Army is the 3d Infantry (The Old Guard), which was originally organized in 1784 as the First American Regiment. However, the oldest units in the entire Army, dating back to 1636, belong to the Massachusetts Army National Guard.