Besides grilling and boiling, Wikipedia lists steaming as a cooking method for hot-dog sausages.
Steaming heats the sausage with moist air rather than placing it directly in boiling water or on a hot cooking surface. It is especially common at concession stands and food-service operations because multiple sausages can be kept hot and warmed evenly without splitting as readily as they may in aggressively boiling water.
Hot dogs are usually precooked, so these methods primarily reheat them rather than cook raw meat from scratch. Steaming can preserve a plump texture and lets the sausage’s seasonings remain concentrated instead of transferring flavor to cooking water.
Wikipedia also describes other possible methods, including broiling, pan-frying, and air-frying. Grilling can add browned or lightly charred flavors, while steaming produces a softer exterior. The cooking method is separate from the style of service: the same steamed sausage may be placed in a bun and finished with mustard, relish, onions, chili, or other regional toppings.