The hospital would usually make sure that your mother has enough of her prescription medication before discharging her.
The pharmacist cannot deliver to a neighbour, they have to deliver to the patient's address.
The pharmacist also has a duty of care to make sure the medication they prescribe is correct. It is not unusual for the same medication to have different names (brand names, generic names etc) and they should be concerned that if your mother is given x branded apple by the hospital and they give her x branded pear (but it is the same) she might take twice as much as she should.
Equally they will be concerned that the hospital could prescribe your mother a medication that will have a severe reaction if she takes one of her old prescription medicines as well.
I think the pharmacy is behaving properly and responsibly. The hospital will (should) be aware of all her prescribed medicines - phone the ward sister to put your mind at rest and ensure she leaves with adequate supplies.