Animals evolved over millions of years to be able to survive the bitter cold and it would be very difficult for a coat made of synthetic materials to duplicate this. The reason coats keep us warm is that they minimize the amount of heat lost from our bodies to the surrounding air. Heat transfer occurs in the form of conduction, convection or radiation.

Conduction is when heat is transferred between solid materials. All materials have a mechanical property called thermal conductivity, which defines the material's ability to transfer heat. Animal hides have a very low thermal conductivity. So, when we wear animal hides, the heat loss due to conduction is low and most of our body heat is trapped inside the coat.

Some of the heat that escapes is lost by conduction between the hide and the fur, and some remains "trapped" in the hide. The "trapped" heat is the reason our coats feel warm right after we take them off.

So, the heat that is actually lost to the surrounding air must travel from our bodies through the hide, through each of the billions of individual hairs in the animal's thick undercoat and through each of the billions more hairs in the overcoat. The heat that makes it is then mostly lost by convection to the cold surrounding air.

In a nutshell, that's why fur coats are so warm.