The airlines started calling these charges "airport taxes"
I think that the justification for this is that, at the time, the airports were generally state owned businesses. Therefore any charge levied by the airport was a charge levied by the state and therefore is a tax.
In some places (France for example) the state still decides on the charges at each airport and these need to be paid to the state - which means that they are a tax. Sometimes charges are paid to the airport and then passed straight on to the givernment (security charges are an example of this) - this is still therefore a tax as it is a payment to the government for services.
And most airlines now classify these as "Taxes and Charges" or something similar so that the tax amount is mixed in with fuel surcharges, wheelchair levies, APD etc.
The main thing is not to worry yourself with how your fare is displayed but to look at the overall price that you haev paid.