Isn't this example different case from the 3D example later on because we rotate this in the same axis for both rotations it seems

Yeah, I agree. For 2D rotation, we are rotating along the normal axis, which cannot be seen from 2D space. but for 3D rotation we are rotating along x, y or z, which are part of the 3D space.

It's interesting how people's real life experiences affect our definition of these transformation. We all live in a 3D space so it's very uncommon for people to think about rotating a 2D image along u or v axis (or x, y axis) because such thing doesn't existing in real life.