How does it not? People pay their taxes, income, sales and use tax, gasoline, whatever form their local government uses to fund roads, and their fair share of the road use comes out of their pocket. So, yes, driving your own car IS actually paying your own way.
Do people not pay their taxes for public transport too?
You had quite specific numbers for the amount of the journey cost that the user pays for public transport, what are the equivalent numbers for drivers?
My intuition is road and parking projects are so spectacularly expensive I doubt drivers are paying their full costs via petrol excise and registration fees.
Obviously it's different in every city, but if we're going to have a conversation about the merits of subsidizing (or not) public transport then we should probably have figures for comparison on privately driven cars.