You only need to set the store name for an item, if that is of help the person doing the shopping – the field is completely optional and can be skipped. However, if you set it, you can easily filter the list only to show items to be purchased at a particular store by just clicking the store name label on the shopping list. Click it again to un-filter the list.
After you’ve bought items during a shopping trip, the trip itself will be marked with the store associated with the location reported by your phone. Normally, the store marking of a shopping trip is automatic, but you may need to specify a location to be a store the first couple of times you shop there.
Note that the trip store marking is totally independent of the store marking on individual items.