It was never a dragon Type Pokemon, there were no strong Dragon type moves in Gen I so it would have been to hard to beat if it only was weak to Dragon. Many many Pokemon have the ability to learn moves of Types they are not, in fact all Pokemon can. It has the appearance of a Dragon-type but is not one, Charizard(Fire-Flying) is also that way.
I know this was already answered, and speed's was a good answer. However, it wasn't sarcastic enough for my liking. So to answer your question, another question:
How can Pidgey be a Normal/Flying type if it can learn Dragon type moves?