The material is genuinely distinctive. Instead of the mineral tanning used for most leather, chamois is made by working fish or marine oils into the skin, which oxidise and bind to the fibres. The result absorbs many times its own weight in water and dries a surface without leaving marks, which is why it became standard for polishing.
The original hides were chamois skin, and the name carried across into English, usually pronounced shammy. Demand long ago outran any supply of alpine goat-antelope. Modern chamois leather is made overwhelmingly from the flesh split of sheepskin, oil-tanned by the same method.
Two confusions persist. One is the assumption that the product is now entirely synthetic: real oil-tanned sheepskin chamois is still widely sold, while PVA sponge cloths marketed as synthetic chamois are a separate material. The other is the animal itself, which is not a goat but a relative of the mountain goat and the serows.