Asthma and Eczema are both results of allergic reactions. Allergic reactions are induced by several stimulants. It varies from one individual to another.
One can have one or more stimulants. Those identified as for your case is Exercise, should be avoided as much as possible and look out for other stimulants as well.
The allergic reaction involved in asthma is hereditary. In early stages of life especially less than 6 years, the similar symptomatic presentationcannot be diagnosed as Asthma as yet. It may be the explanation of your diagnoses that you still have doubt in.
You may need to see a Physician for proper review.