If anything the joke is its a cute doofy rabbit being called a muscleman. Now THAT i've seen lots of fan art of.I legit can't think of any reason why azumarill is meant to have huge power (and, by pure chance, was actually thinking about it earlier today). Google searching it yields no real answers - even google image searching for stuff like "azumarill mochi" or "azumarill rabbit on the moon" in japanese doesn't net a single piece of fan art, which gives me the feeling that it's not meant to be that.
Of course the real question is who all on staff liked Azumarill enough to give it such a significant buff before many other pokemon.
Also the fly question going on reminds me: why can Garchomp not learn fly?
Several of its pokedex entries say it flies around. Its design is partially based on a jet. The anime shows it flying all the time. And yet, can not learn Fly.
The fact it will often be shown flying in other media makes it even sillier than Gyarados not getting it.