No, even my dog has hair, not fur. Breeds like terriers, and poodles have hair that will continue to grow if it's not cut, and don't shed. German Shepherds, beagles, and Labs have fur which sheds and regrows. You can be glad that humans don't have fur, or we'd be shedding all over the place, too!
No. Hair and fur are not the same thing. If you, for example, ruffle the coat of what they call a "double-coated" dog, you will find "outer hair" which is actually hair, like human hair. The "under coat" is FUR, which is finer than hair, and which humans don't have. A "single-coated" dog only has hair. The fur is designed to trap air and provide insulation against cold and to some extent against heat. Hair can do this also, if the animal can "fluff" the hair up. That's why humans get "goose bumps"; we're trying to fluff up our body hair to say warm, but we don't have enough.
We have HAIR, some more than others. There is a very rare condition, largely linked to hispanic genetics, called hypertrichosis in which thick fur like hair grows over the entire body. Most likely the origin, at least in part, of the werewolf myths. Ive known a couple of different, unrelated men, who had LOTS of thick dark hair on chest, stomach and all over their backs and also had very dark beard shadows on their faces.