If I understand it a bit the standard answer for this is: the pushing/holding apart of fermions and the pulling/holding together of bosons is just a result of symmetrization requirement (Griffths' term). But if it does force particles to move or be in a certain place, how come it is not a force? Is this just a way to say it is not the result of any of the four fundamental interactions?
It just means that the exchange force is not a classical force like between spinless classical particles. It is a purely quantum effect. It disappears when conditions become more and more classical.