Is it permissible to benefit from a commercial discount to purchase an item for another person, while keeping the invoice and the item in the name of the discount holder without the shop's knowledge?

There is no harm in purchasing what a parent or relative desires using a discount available to you, if the discount is given to a general category of buyers and not for a personal consideration. Some stores grant a discount card as a gift or to those who have purchased above a certain amount to encourage further purchases from the same store. There is no difference between buying for oneself or for others, and no harm results from this to anyone. For those who wish to be cautious, they may ask the store owners about the use of the discount for others.