Can someone be allergic to aspirin and what would be some allergic symptoms?

Yes: The answer is swelling and/or hives. This is different from aspirin-sensitive asthma or aspirin-exacerbated respiratory disease which is triggered by nearly all nsaids

Aspirin allergy: Aspirin can cause hives or anaphylaxis and it can cause unusual reactions including asthma wheezing and rhinitis and conjunctivitis similar to hay fever in people who are sensitive. People with asthma and nasal polyps are prone to this reaction which can be severe. Some also react to a similar group of drugs called nsaids which includes ibuprofen. Desensitization by an allergist is possible.