That is me. As a child.
I had allergies to stuff in the air. ie: pollen, dust, pets.
Plus I had childhood Asthma and still have it.
I was just like your child.
Yes, an inhaler will help thusly.
An inhaler is used because, Asthma or triggers, makes the bronchial airways constricted. It gets constricted because it is inflamed.
And if constricted, a person cannot breathe, normally.
As a child, my parents got air purifiers, for my room and all the rooms in the house. And I did not have any carpeting, in my room.
And though we had pets, it was kept outdoors.
NOT indoors... because, pet hair and dander... WILL BE stuck to all the surfaces in a home. And a person cannot clean it all entirely.
It can even be on a person's clothing, and you will not know it because pollen/dander/hair can be so small that you cannot see it with the naked eye.
Pet hair/dander will affect a person in these conditions.
For me, it was also certain pets and cats. Not all breeds.
So this is not uncommon.
However, Poodles and Airedales, have "hair" not fur. So this is less allergenic. But cats, and their dander can be very allergenic.
Know that.
You have to manage her condition/allergy to pets/asthma... or it will get worse.
Asthma... can be very serious. People die from it.
And, every person has a different trigger for it. What may trigger my Asthma may be different from the next person. So you have to know, what triggers your child's condition.
And you do need to follow directions for her medications. And use it.
If a person cannot breathe or has an asthmatic attack... that means they cannot breathe and/or are not getting enough oxygen into their bloodstream either. Not only about their lungs.
You need to do, what the Doctor recommends.