This was brought up on here about a year ago as I recall.
Technically there is no such thing in the US army.
In practice, there is. Typically if a second set of eyes "confirm" the kill or if the rest of the unit and/or lawyers move in when the shooting stops and start handing out cash, they'll take a count. In some cases, only the bad guys are counted, in others they count the total body count (like with the lawyers). All of this shit goes out on the net or is picked up on by that buddy you have in MI. These body counts are for more than just to confirm kills (and it doesn't necessarily say who killed 'em).
And like was mentioned, the orders for an award for action resulting in enumerated enemy deaths sort of serves as one. Arcom? Shit, I saw one handed out to a fat supply guy simply because he didn't fuck up his job at NTC. Jeezuspleezus.
When we got a new 1SG prior to unit deployment, he came from the ranger batt. down the street and they'd already deployed. More than once (but they have very short deployments comparatively). Anyway, he had a Q&A before we got down to business. He was short, so he allowed the height question, 4'11" (barely 100lbs, all gristle and bone, he was a tough SOB). And the confirmed kill question, to which he answered : forty (something). This was in the role of sniper as I understand it, but not limited to that. There were others.
The whole time I was in the army, it was never discussed. Not once. The only time it ever came up was when that kid asked the 1SG that. In SDM school, we didn't teach soldiers how to count kills, we taught 'em kinetic action, move through and take out the enemy, fuck counting 'em. They don't teach it in sniper school either.
That's why technically, it's not a thing.
Confirmed kills on war material, that's a different story. We do historically keep track of the number of planes or tanks or whatever that are taken out and display them in some fashion as bragging rights for the crew. This was done widely in WW2, I don't think it's as big of a deal but maybe it depends on the service and/or unit.