when you kill a skeleton horse isn't the same you don't get the advancement "monster hunter" but you kill a zombie horse you can get it

normally this advancement can be achieved by killing the following mobs:

.Blue spider
.Spider
.Zombie Pigmen
.Enderman
.Polar bear
.Blaze
.Creeper
.Evoker
.Ghast
.Guardian
.Husk
.Magma Cube
.Shulker
.Silverfish
.Skeleton
.Slime
.Stray
.Vindicator
.Witch
.Wither skeleton
.Zombie
.Zombie Villager

moreover you can see in the screenshot n ° 1 the zombie horse is invisible and makes appear its shadow on the ground you will have more information on this issue MC-124147

the zombie horse isn't a hostile mobs.