Allergy to Aspirin is more likely an allergic reaction to NSAIDS, which are anti-inflammatory drugs. It is a pretty common to experience a reaction while taking Aspirin.
Symptoms to watch out for include hives, itchy skin, swelling in the face and tongue, and shortness of breath. Although rare, anaphylaxis can occur as well.
Treatments include avoiding any medications with NSAIDS, allergy injections could also be a solution but you should ask your primary doctor first or see an allergist.