I figure the operational size for both animals was pretty much the same ~ because anything beyond human scale would put them beyond "control".
It's only in modern industrial times that we've gotten into using machinery that dwarfs us, and that's only because we build in OFF SWITCHES. Elephants and horses have an off switch but it's buried under feet of flesh and gone. You need a spear to reach it! Last thing you want is an animal you can't turn off ~ like a bull African elephant rampaging in must.
From Wiki....but the sources look legit.
15 - 16 hands = 60-64 inches tall = 5-6 feet tall at the withers or top of the shoulder.
Elephant = Asian male elephant around 11 feet tall. African elephant 10 to 13 feet at the shoulder.
Nobody has ever seen any evidence (armor or skeleton) of a horse as large as an elephant.