I speak as an asthmatic, not someone who knows about medical requirements.
Firstly, you don't have to have had a classic asthma attack to have asthma. Other symptoms include breathlessness and coughing especially night-time coughing.
Secondly, you say nothing about why your GP thinks you have asthma and you don't. What diagnosis was done? In my case I was prescribed a peak flow meter and had a classic saw-tooth pattern when I recorded morning and evening values. Mine was a classic way for an adult to develop it - I got a nasty virus which set of a cough which never went until I was on a steroid (brown) inhaler. That nailed it. I almost never take a reliever (blur) inhaler.
In a friends case the asthma nurse at his GP did spirometry and repeated it 20 minutes after giving him a dose of a reliever. He had bad asthma as a child, effective treatment was almost nil (we are talking about 50 years ago), he grew out of it but as with about 1/3 of such people he aged back into it. Being a man he was ignoring things like coughing and wheezing, but last time I saw him he found the improvement in his breathing following the right (steroid) medication a revalation.
So, what is your history?
Thirdly, what kind of inhaler have you got (blue I suspect) and how come the pharmacy are delivering them even though it seems you are not requesting them via a repeat prescription form? For heavens sake don't let that continue to happen - if nothing else it's a ridiculous waste of money.
There's a lot of information on the Internet about asthma though as always some of it is wrong. Suggest you browse the following website and see what they have to say:
Asthma UK - Homepage
I don't know what age you can choose your GP at, but at 17 I'd say you are either there, or you will be when you are 18. Obviously for a small child the parents choose the GP but you are no longer that, and in your situation I think you should see the asthma nurse. They are specialists in asthma and often know more than the GP does. My luck was a good GP who recognised firstly how to diagnose it and secondly that the treatment of choice for the vast majority of asthmatics is a steroid inhaler.