Yes, it sounds like an allergy and she should definitely avoid cashews. The oil is probably processed in such a way that the protein causing the allergy is removed. Many children with peanut allergies can safely have things cooked in peanut oil. I would cashew oil is the same.
Allergies can vary in degree. While a raw, whole cashew may be too much, having it as an ingredient can still be ok. My son can eat brownies, cakes, cookies etc that have eggs in them, but he cannot eat eggs any other way (scrambled, hard boiled, etc). The process of cooking them removes or alters the protein causing the allergy.
Cashews have a strong cross-reaction with pistachios. I have also hear they may cross react with mangoes. So, if your daughter is eating either of those, I would also check for a reaction and ask if her throat bothers her.
Finally, allergies can get worse with repeated exposure, so if whole cashews are a problem, she really needs to stop eating them until you talk to a doctor.