It's not true that 'fur is hair'.

Let me quote you something:

"the most general distinction is that a dense coat of fine fibers is usually called fur. A less dense coat of thicker fibers like what you find on a human or a horse or a pig is usually called hair. And then, just to complicate things further, a very fine, very dense coat of fibers is often called wool, as with sheep, alpaca, and muskoxen. On the other hand, the animal with the finest, densest coat of all is probably the sea otter, but I’ve never heard their fur called otter wool – so go figure." via 'askanaturalist' website.