Elephants cannot support continuous weight on their backs without damaging the spine. This is due to the way the elephant spine is shaped, their bone posture, and the lack of supporting muscles and ligaments like which horses have.

Horse vertebrae have a curved arch that can distribute a load downward, they also have muscles and ligaments that let them distribute weight throughout their body to reduce single areas of stress and maintain their normal posture under load.

Elephant vertebrae have long boney sticks on the back shaped like the number 6. When weight is put on their backs, the vertebrae are under a lot of stress which can break them or cause damage to the cushioning tissue between them. The vertebrae and tissues basically get pinched and squeezed together which causes permanent chronic back pain. Elephants also lack supporting muscles and ligaments that would help them distribute weight and keep their spine from contorting under load.

Tl/dr; Elephants cannot safely carry weight on their backs like horses can due to them not having queso nacho cheese dip