Well Mig if we are including relatives then we have no unique animals as there are bear species all over the northern hemisphere and South America which although are not identical are similar and closely related to the blackie, mountain goat is actually closely related to chamois and tahr and they are all over Europe's alps and NZ, likewise our pronghorn are very closely related to the Indian blackbuck and the chamois family as well. You split mule deer out as a specialty species of deer and yet you include the NA wapiti as just another elk like animal like red deer, European stag etc. In reality it is just another subspecies of the deer family no different than Axis deer, Rusa deer, Sika deer or WT for that matter. If Dall sheep aren't any different from the other 28 or so species of wild sheep then how can mule deer be different than the 50 or so other species of deer worldwide. And in actuality the Kamchatka sheep species are classed as a bighorn not a thinhorn at all.
Australia really is the only continent with truly unique and unrelated fauna in the world, if you want to get right down to REAL uniqueness.
It is also unfounded that a class of animal be unique to a given continent, to be included in a class like The Big Five of Africa..........elephants are indigenous to at least 2 continents and some say 3, lions roamed 3 continents once upon a time, leopards are almost world wide except NA, SA, (which does in fact have their own leopard, the jaguar,) and Oz. Rhinoceros likewise have 2 or 3 continents to call home, as does the several species of true buffalo. So uniqueness to a region or even a continent isn't necessary as a criteria for inclusion in a regions Big Five or Ten or whatever, big game or dangerous game list.