El elefante tiene el cuero muy duro.

English Translation

The elephant’s skin is very tough.

Wouldn’t you use “la piel” for the skin of a live animal?

To be thick skinned (tener el cuero duro).

Elephants do have ‘thick skin’.

Animal hide can also be said as “leather” and both “cuero” and “piel” both fit here.

I’ll say the original sentence is correct. (good question!)