Are all tailless cats Manx?

No, not all tailless cats are Manx. While the Manx breed is famous for its taillessness due to a genetic mutation, other breeds (and individual cats) can also be tailless or have short tails. 

This condition can occur due to different genetic factors or physical injuries. Some other breeds, such as the American Bobtail and the Japanese Bobtail, also naturally have short tails but are distinct from the Manx breed.