Since her father is the actual rightful leader, Korra does have royal blood and is the new princess of the water tribe. Although she is the avatar, she naturally has more special privileges cast upon her.....so yeah, she's a princess.
I'm so against this because it makes Korra Mary Suish in a sense..... ughhhh
I doubt they'd let the Avatar, who is meant to be neutral in these things in order to keep the balance, become the leader of one of the four nations. It's possible she'll be offered and decline for that reason, or that someone else will end up in charge or even that they will change the system entirely, the way they did in Republic City.