Is my baby allergic to breast milk?

No. (probably not). It is very rare for babies to have intolerance to breast milk. On rare occasions babies may be intolerant to a food they are exposed to in their mother's diet through breast milk. Allergy skin tests can help determine if this may be the case. Common symptoms of this may include rash/eczema or even vomiting/spitting up. It should be noted that most colic is not due to food allergy.
No. Infants seldom are allergic to breastmilk. That would require a true milk-protein allergy and, fortunately, those are rare. Symptoms are vomiting, diarrhea and either weight loss or poor weight gain. Your child needs 3 of these symptoms. The diagnosis is usually made by taking your child off all dairy products for 6 weeks, then reintroducing dairy to see how the baby responds.
No. Yes, it is possible to be allergic to breast milk. (it is possible to be allergic to any food item.) however, babies are naturally designed to drink breastmilk as their source of nutrition, so it is quite rare for a baby to be allergic to it. The parents should visit the doctor, with the baby, and investigate further, before deciding that there is a breastmilk allergy.
No. Your baby may have an allergy to something that you are eating that is coming across in the breast milk but allergies to breast milk, itself, don't happen.