As I understand it the European Elk is related to our Moose. The European Red Deer is related to our Elk (Wapiti). The European Fallow Deer (what Robin Hood hunted) is related to our White Tailed Deer.
Not quite. The two larger animals are essentially identical in the two continents, with the only real difference being the names.
American whitetail and mule deer are completely different species from the European fallow deer.