Allergy to Nightshade vegetables is actually quite common. My DD has a topical reaction like the one you described to eggplant.
Keep an eye on her when she eats other veg from the family, including paprika and cayenne pepper, eggplant, capsicum and even potatoes. She may have reactions to other, she may not. DD doesn't have any other allergy or reaction to anything at all.

Make sure you tell her caregivers or day care, grandparents etc so they don't give her tomatoes.

HTH