As a general rule, if something reaches the throat through the nose or mouth, it is prohibited. Thus the use of inhalers is prohibited whilst fasting. For the person that requires it, he or she has been granted a concession not to fast due to illness.
In summary, the inhaler does break the fast and the day should be made up if one is able to make up days. If one has chronic asthma and cannot make days up, it will be sufficient to pay fidya.