It seems to me like the issue is not that she's too sensitive. It's that she doesn't know how to deal with her sensitive feelings. Being sensitive is not a flaw. Think of the people you've met in your life who instinctively knew you were having a hard day, or sad, or struggling. Think of the people who knew you needed a hug, or someone to listen. Think of the kind, quiet, thoughtful people who sense what's going on around them. We need them, just like we need the non-stop action go-getter take-charge rough-and-tough people.
The problem happens when the sensitive people don't know how to use their gift. One time I remember a family member was in the hospital with a serious illness. He was on the long road to recovery, and he had a friend who came and visited every day. But the friend just sat silently by the bedside, with tears in his eyes, like he was keeping vigil at a funeral or something. It was so depressing, to the patient and everyone. We finally talked to the friend and said we appreciated that he sat by his friend's side, but the sadness wasn't helping. He explained that he was the sensitive type. We said we understood that, but it would be more helpful if he turned his sensitivity into positive action. Thankfully he got it, and next time he brought a great book and read it aloud to the patient. He started bringing music, got the hospital therapy dog to visit, and turned his sensitivity into helpfulness. The patient recovered and the visits were no longer melancholy, but positive.
So, it's just my unprofessional opinion that the answer lies with knowing what to do, and it won't always involve ignoring the feelings or just brushing them off. Instead, it sounds as though your daughter needs to learn what she can do. If a kid says something hurtful, perhaps she can learn how to respond with kindness, or say "stop", or tell an adult if it's a very damaging thing that was said. If she is upset by a friend, should she call you, or should she speak up and talk things over? Or should she physically remove herself from the situation, and do some calm breathing, or recite a verse or saying that means something to her.
I guess I mean: action, rather than passive behavior, might help her.