The chemical structure of caffeine resembles that of a common asthma medication, how, and what does this mean?

Biochemistry: That means that the asthma medication is actually a stimulant that also has activity in the smooth muscle of the bronchial tubes to allow them to relax and not be so constricted.

Not much: The asthma medications that coffee mimic fell out of favor decades ago when modern treatments became available. In order to get an equivalent benefit to the old aminophylline products ( the chemical cousins) you would need to take so much that the side effects of hyper alertness, jitters, etc. would be prohibitive. For the most part it is just doesn't worsen asthma.