Is it still considered asthma even though the child doesn't have any problems wheezing or breathing unless sick?

Possibly... Asthma is recurrent episodes of bronchospasm and airway inflammation. It is possible for these episodes to occur only with infections.
Astma is a disease. Asthma is a disease that affects he bronchial tree (it narrows) and that is what causes the wheezing. This can occur only when your child has an infection and therefore although it is considered mild it still is asthma and he should be evaluated because it may worsen as he grows. He may not need any treatment but should be followed.
Asthma can be mild. Asthma is defined as reversible bronchoconstrictrion, this can be mild that it can only be detected by lung function test. Some asthma patients only realize their condition after a puff of albuterol, as they get use to a mildly reduce pulmonary capacity.