Something Amyss said:
It's not that Americans don't know how royalty works, it's that George Lucas doesn't know how royalty works.
He wanted a democratically elected leader for his story, but since he was using fantasy tropes to create his universe he decided he needed to have a princess somewhere, just like he needed a wise old man and an idealistic farm boy with a magic sword. Of course, this technically means that you have a monarchy fighting against a dictatorship in A New Hope, and a 14 year old elected into office in Phantom Menace.
Dare I say, a better writer might have written around it.
Quite easily. All the writer would have to do is make the title part of their cultural heritage, while still making the character democratically elected. Or, alternatively, certain worlds have monarchies, but they still have representatives in the space-senate. Basically just do this:
Of course, Lucas didn't do that. He seemed to make Leia a hereditary princess when she should have been elected, since she's fighting for representation. Meanwhile, Padme is evidently a 14 year old elected official, when she should have been a hereditary figurehead. You can have a teenage princess if it's hereditary, but it's ridiculous to suggest that she was voted into office.
Again, we're thinking way more about this way more then he did. He just wanted fill out as many spaces on his Heroes Journey bingo card as possible, so every girls got to be a princess.