So far as I know, horses are limited to one skeleton, period. They aren't like Races, where you can define a different skeleton, unless that's something new that's been added in 1.150+.
So you can make an "elephant" skeleton, but the engine won't use it.
More importantly, even if that problem could be gotten around, it would still use horse animations, etc.
About the only way around this is to make a new "horse" skeleton that has the "elephant" bones and keeps them buried underground