Board Thread:General Discussion/@comment-25086632-20151123071037/@comment-26397825-20151212095655

They have a Council in Atlas, which kind of negates the idea of a monarchy. Plus, it's a military state, and has been implied to be run by Atlas Academy (hence, it is called Atlas and in one of the WoR's actually states that the Kingdom was Mantle, which is now defunct). Atlas Academy is run by Ironwood, who is also the General of the Atlas Military.

I don't think this leaves much room for there to be a monarchy. It's far more likely the title of princess is a moniker meant to describe their rich lifestyle, and pampered upbringing (well, other's view it as pampered. The people living those lives? Not so much), not a literal description or title. It's a nickname, one that Weiss isn't actually all that fond of. Case in point:

Roman: (to Ruby, Blake and Yang) Girls (to Weiss) ...Ice Princess.

Weiss: Hey! (in a 'Hey! Don't call me that' manner)