I guess for me it just feels like a slightly twee and even patronising new-coined name, like with Painted Dog - except with Moon Bear, I've only ever really seen it used by animal welfare charities, who by definition are after the public's money and sympathy (and using the name no doubt for the exact reason you give, @Zia). I'm sure I'll grow used to it in time, as it seems to be spreading. It's pretty irrational, to be fair, given I've no problem with Sun Bear (though, actually, maybe the artificial neatness of that pair of names is part of my problem! I'm quite sure I'd feel the same with White and Black Rhino if those names didn't pre-date me).
Though I do agree that it would never cross my mind that it was referring to Sloth Bear!
Doesn't the old version of their scientific name (Selenarctos thibetanus) literally mean “moon bear of Tibet" ? I'm sure I have read that somewhere.