A couple of ideas:
1) Marines. For many decades, the American armed forces had a virtually unique capability for large-scale ampiibious assaults against on-the-beach defenses. Most nations' "marines" could only land against no resistance or token resistance, and fought once established on land. There were minor exceptions-- some armies had a few battalions worth of "commando" or "special forces" units that might conduct such operations. During World War II, the Brits and Canadians both had landing forces in in division strength, but this was a short-term capability; only the Americans maintained the training, naval capability, and troops for such operations, later used extensively in Korea and to a lesser level in Vietnam.
2) Air Cavalry. The game currently offers helicopter gunships, but the United States has consistently maintained a strong air-infantry capability since the second world war, originally with paratroopers-- that remain-- but since Viet Nam also with a strong helicopter-based air cavalry component. In the second world war, the major powers on both sides had some gliders and helicopters. The Germans pretty much gave up paratroopers after the Battle of Crete (their "paratroopers" such as the Herman Goering division had neither chutes nor chute training, though they did maintain some special forces units). The capability for large scale airborne operations after the end of the Second World War was pretty much limited to the Americans, though the Brits retained some limited capability, used in the Battle for the Falklands.
These could be treated as unique units, but the limiting factor was really one of cost and time. For real marines, paratroopers, or air cavalry, a lot of extra training is required, and some very expensive naval or air force infrastructure (e.g., combat landing ships) are required. If marines and air cavalry were simply excellent-class infantry with the additional movement/landing capability that were VERY, VERY expensive for their combat capability, they'd only be built by relatively rich civilizations that needed their specific capability.
For a truly unique American unit, it's worth considering that Americans fielded troops with the Kentucky Long Rifle-- low rate of fire but VERY accurate-- and, in a similar vein, unlike our European counterparts, American musketmen were drilled to AIM their muskets. European infantry were trained to look away from their targets when they fired. This was safer with some muskets, and contributed to a higher rate of fire, but American fire was more accurate and, hence, had a somewhat longer effective range. Thus, one might introduce Early American Infantry in place of musketmen, that would actually be a mix of musketmen and riflemen. Their "special capability" would be "first shot," with, perhaps, a higher combat value on the first shot. The former would reflect the longer range; the latter the value of aimed fire. Legitimately, though, after the first exchange, they'd be no better than normal musketmen, and maybe even weaker, since American troops had a lower rate of fire and were not as well trained for melee with bayonets.