My sister has food allergies - milk and egg specifically that cause painful acne looking allergy pimples. One way to check out if its an allergy is to keep a food diary. It was only when she looked at her diary that she was able to make the connection and insist on seeing an allergist who confirmed it. Now she cannot have foods made with either egg or milk.
There are alternatives to using milk or eggs - we have sorbet or Tofutti instead of ice cream, use non-egg based egg replacement (stupid people use egg in some of the common egg replacement products WITHOUT listing egg on the incredient lists) in baking goods and use rice/almond/hemp milk. Be careful with soy though because many people with milk allergies tend to have soy sensitivities. Also read labels carefully - its not always clearly marked as containing eggs and milk as ingredients.