I would not consider her to have sensitive skin. But that doesn't mean she couldn't be sensitive to one thing. Is lemon more acidic than oranges? She wolfs down baby oranges with no problems. (One of my other grandkids loved oranges but could not eat them until he was over 2 because even if you changed his diaper as soon as he pooped, his butt was red and sore enough he cried.)
I have no idea what baba ghanoush is. Probably dil made it herself. She assumed it's probably the eggplant, which I think is the main ingredient as well as something baby has not had before.
The red skin, but no issues from eating (no coughing, vomiting, diarrhea, sore belly, etc.) has us both thinking more a sensitivity than an allergy. However, I don't have experience with babies/toddlers with food allergies.
BUT, neither she nor I have any experience with baby's with food allergies.