They are analyzing models trained on classification tasks. At the end of the day, classification is about (a) engineering features that separate the classes and (b) finding a way to represent the boundary. It's not surprising to me that they would find these models can be described using a small number of dimensions and that they would observe similar structure across classification problems. The number of dimensions needed is basically a function of the number of classes. Embeddings in 1 dimension can linearly separate 2 classes, 2 dimensions can linearly separate 4 classes, 3 dimensions can linearly separate 8 classes, etc.