FashionMNIST inherits raw_folder and processed_folder from MNIST without a change. If you download MNIST and then try to download FashionMNIST in the same root folder, it'll say it's already downloaded while, in fact, it's not. After that FashionMNIST just uses MNIST data.
For a quick fix, make them as properties that depend on __class__.__name__. A better change would create parent folder with the class name. The user has no idea to what dataset 'raw' and 'processed' folders belong if he/she looks at his './data' dir unless he goes through all dataset implementations in torchvision.