I used a vector mask to mask an object from its background in Photoshop. I know I can apply a general feather in the mask properties. However, the object is more fuzzy around the edges in some places. If I was using a layer mask, I would use the brush with a low hardness to those zones to make them more fuzzy on the sides.
Is something similar possible with vector masks?
No, not at current and it seems unlikely that something like this should be implemented for vector objects for the time being.
But you can load the Path as a Selection and use that as a Layer Mask and edit it further.