Earlier quoted context omitted.
Republic and democracy refer to two different axis. Republic is about how the vertex of government is structured. Democracy is about where the executive/legislative power comes from. The US is both, you could argue that Canada is not a republic as they have a recognized queen, but it would be a stretch as she has essentially zero power there.
They said they elect their leaders democratically which is simply not true no matter how you want to phrase it. The president is elected by an electoral college and not a democratic vote.
There's an old game, Anachronox, that does a wonderful job of making fun of this when the player is on the planet Democratus. Worth a play if you haven't.