Presumably in a galaxy where a 14-year-old can be elected queen, it's fine for 19-year-olds to be senators. Maybe being a senator is hereditary - after all, according to the movies her adopted father was one.

Transcript

GM: Princess, you're marched into the presence of the black-armoured figure.
Leia: Aren't you a little tall to be a clone trooper?
Darth Vader: And you are too young to be a Senator. Hmmm.
Leia: Old enough to kick your shiny plastic arse.
Darth Vader: You shall address me as Lord Vader.
Leia: You murdered Uncle Antilles! You ain't gettin' no information out of me either!
Darth Vader: So you do have a copy of the Peace Moon plans?
Leia: I'm ready to die for the Rebellion, "Lord" Vader.
{beat}
Darth Vader: Did you just air-quote me?
Leia: "No".
Darth Vader: Take her "away".