Benjamin Franklin's stove was not the first, and there a number of much earlier designs. Which of these was the first stove depends on what is meant by the word stove. There are masonry and ceramic stoves that were older in Europe, and the Kang bed stove in China. But these did not provide the same utility as the Franklin stove.
Today, the original Franklin stove would probably not be called a stove either, but rather a fireplace insert. The stoves sold as Franklin stoves are not the same design, and were developed by people who came after Franklin.