A Møøse once bit my sister … No realli!
But sadly, I cannot answer your question.
Yes, they are. Or so it seems.
Wiki is your friend. 
As the article says, they’re all the same species, Alces Alces, but different subspecies. I don’t know what differences exist or how significant they are.
Yeah, that wiki thingy seems to say it all, but I really liked the Newfoundland warning sign… snerk…
Sadly, we all look identical to them, too.
I’m kind of curious why they would be the exact same species as the European continent has been separated from us from some time now. As an example there are North American Beavers, and European beavers, but the European beavers are distinctly different.
Perhaps one of our evolutionary biologists may have a definite answer for this. In general, what I’ve noted is that megafauna – the largest, dominant animals in various econiches – tend to be Holarctic in distribution; that is, they exist in the north-of-the-Sonoran-Desert-and-Alps/Tien Shan/Gobi Desert ecological regions in both Old and New World. Smaller animals tend to be Palearctic and Nearctic, with related species inhabiting the two regions separated by the Atlantic and Bering.
In particular, this seems to hold true for those with tundra/taiga ranges. Contemplate the mammoths, all now extinct. The woolly, Mammuthus primigenius, was Holarctic in range; the various non-shaggy mammoths were confined to either Nearctic (Colombian and imperial) or Palearctic (southern and steppe). Likewise, the “cattle” econiche has the muskox (Ovibos) as originally Holarctic (though today confined to Alaska and northern Canada). Farther south, the aurochs (Bos primigenius and its descendents were exclusively Old World, while Bison exhibits two species, one American and one European (the wisent).
From a non-expert, one thought is that, in general, larger animals have longer generation times. So for a given period of time that two areas have been physically seperated, the smaller animal will have had more generations, therefore more time to genetically drift apart into seperate species.
Also, and I’m not sure this really applies to European vs American mooses, larger animals can travel over distances and physical barriers that smaller animals can’t, allowing enough genetic mixing between populations to keep them the same species.
They’re often not shaved! <badum-dum> Thanks, I’ll be here all week.
Otherwise, yeah, Poly and that oak guy have nailed it.
In addition to what Polycarp and Quercus said, it is likely that there was little environmental change to cause any evolitionary pressure to move far from the original common genetic heritage. Both European and North American moose/elk inhabit near-identical environments, eat the same type of foods, and are hunted by similar predators.