This, indeed, can be a case of dog allergy.
Dogs have a substance, protein excreted in their saliva that is very potent allergen, along with other proteins-allergens in dander (dead skin that is shed), and urine.
Symptoms of dog allergy could range from red, itchy, or swollen eyes, red skin lesions, nasal congestion with runny nose, sneezing, stuffed or itchy ears, post nasal drip, itching and horseness in the throat, coughing and wheezing to frequent bronchitis.
Presence of pats near the child with already diagnosed asthma may lead to worsening asthma and triggering frequent asthma attacks.