Lately I have noticed that a lot of people around me have started to use the name of one tire chain Discont Tire to mean any tire dealer similar to the way when you ask some one if they want a coke and they sure I'll have a 7-up for example. Last week a coworker said he needed to go toDiscont Tire and get a new set of tires but wound up buying them at Sears. So far this seems to be a local thing. As to the owner I either call him the dealer or proprietor.
If they just sell tires, it would be a tire store. But a lot of people buy their tires from a store that is more general; they might go to an auto parts store or a mechanic. Also, in the US it would be "tire store" and in the UK it would be "tyre shop". I don't think "tire shop" would be common in either.