Legolas' and Gimli's contest at Helm's Deep

[The Saucepan Man]

Pre-modern warriors actually had quite a thing about counting kills.

It's not just a pre-modern thing either. In WW1 and WW2 (and in more recent wars too) fighter pilots and tank crews have chalked up the number of "kills" on their respective machines.

It's not really "killing for sport" though. The battles fought in the War of the Ring were visited upon the free peoples. Like WW2, it was a war that had to be fought, for the sake of freedom from oppression. Legolas and Gimli at Helm's Deep, in common with all the Free Peoples during the War of the Ring, had no choice but to kill those who were attacking them. In a spirit of comradeship, and (as others have said) as a way of trying to divert their attention from the seeming hopelessness of their situation, they made a competition out of it. I am sure that Tolkien witnessed similar acts of comradeship and bonding in the trenches during WW1.