My son had (has) asthma. It was accompanied by eczema, colds, sniffles - the gamut of symptoms. It took pediatricians until he was 2 years old to officially diagnose him with asthma. Now, we all knew what it was, he was treated for asthma, but no one wanted to label him with asthma at such a young age thanks to possible insurance complications (like being denied coverage) later in life.
We have had a nebulizer in the house for the past 15 years - got the first one when he was 6 months old and had RSV. Many of his asthma attacks, at first, were bronchial asthma that were off shoots of a plain cold, so we treated the symptoms of the cold and added breathing treatments. As he grew, his asthma would be triggered by allergies, colds, and physical exertion, so we began to carry an inhaler. - the child played soccer for several years in spite of this.
What changed our world, and finally helped me learn to manage his asthma, was taking him to see a pediatric allergist who was also one of only 2 doctors in my area that specialized in pediatric asthma. He introduced knew regimes that significantly helped my son. But this did not happen until he was 2 years old. Also an age when several preventative medications can begin being taken.
So, open a dialog with your pediatrician, do a ton of research on pediatric asthma so you can have an informed discussion with him. I researched treatments and preventive alternatives and with my doctor's "blessing" we tried herbals, and alternate treatments.
If you feel that you are not being heard by your current doctor - find another one, or find a specialist that deals with asthma in young children. A dermatologist can also help with the eczema.
Keep advocating for your child. Asthma is manageable - my son, at 15 now, has outgrown most of it and rarely suffers an attack.
Good Luck
God Bless