Five Guys uses Hebrew national all beef hot dog ( do not know the size),. They call it Kosher style because the way it is cooked (assume in proximity to the cheese... ) is not consistent with Jewish dietary law. Personally I think their hot dog is much better than their hamburger.
Directly from the Five Guys Website.
What is a 'kosher style' hot dog? Is your beef kosher?
We use Hebrew National hotdogs which are kosher. However, we've received feedback in the past from customers who said that although the hotdogs we use are kosher, the way we cook them and serve them is not. As a result, we didn't want to be misleading since we use buns that contain dairy and in most cases the dogs are cooked on the same grill as our burgers, therefore we decided to call them "kosher style". Our beef is not kosher.