Melanie is very confident and self-assured, and cares little to nothing about anyone else's opinion of her. And while she is outgoing to a degree, she's also somewhat antisocial/prickly toward people she doesn't know. Given everything about her attitude shown so far in the game, it's pretty clear she chose to get them because she wanted to, not for anyone else's benefit (though now that she's with the MC, assuming one hasn't closed her path, she does seem to like to show them off for him). I am pretty confident that's the reason Prom gave them to her, to represent that self-assurance and "fuck what anyone else thinks" attitude of hers, which is also supported in her flashing them to the MC in the cafeteria without giving a shit about anyone else there seeing.
The weird puritanical attitude some people here have about piercings (and sometimes also tattoos) has no real basis in reality (not that these games need to pay much attention to reality in the first place). People get piercings, including nipple piercings, for a lot of reasons, but often it's for personal expression or to exercise/demonstrate confidence or individuality. Having nipple piercings does not automatically mean a person is "sex positive" or promiscuous.