Her response to your fear that she might be labeled negatively because of this piercing could be that you have taught her not to let the superficial, baseless stereotyping of others carry any weight with her. She may want to pierce her tongue partly because her "hip mom" has a pierced navel, but I doubt that is her major reason. Notice she is piercing her tongue, a more blatant move than navel piercing, which people do not usually observe in everyday life. I don't think she wants you both to be twins in this regard.
I would calmly make your arguments against the tongue piercing, being careful not to shame or blame her as you explain your disapproval. Have a discussion -- don't lecture her. Then, if you feel that she has argued reasonably for why she could safely have her tongue pierced, you must decide whether supporting this or attempting to forbid it will be in your daughter's best interests and in the best interests of your relationship with her.