It means that the horse and buggy was not yet obsolete.
"Going the way of the horse and buggy" implies becoming outmoded and useless, because cars have replaced horse-drawn carriages. But in 1903, when I was but a wee lad, the horse and buggy was still the prevalent mode of transport. It had not yet been replaced by the "horseless carriage" (cars). As the passage says, cars were just beginning to be seen in Frisco at the time. They were far from displacing horse-drawn carriages for transportation. Hence the last sentence: "In 1903, the horse and buggy was not [yet] going the way of the horse and buggy."