When you use a debit card and process it as credit, it's charging it directly to the bank/credit union who is issuing the card. Some banks/credit unions charge a fee for using it as a credit card, but those fees are charged to the cardholder and not the company.
Debit card - use as credit - I pay a fee from my bank out of my own pocket
Credit card - Target pays a fee to process the card through the payment company
Its for Target's benefit. Gift cards are normally paid for with credit cards, so it would be a no sale for Target if they were used to pay off that same amount on the card.
They would even lose money since they are charged a fee from credit card companies.
So you can pay your bill by credit through your debit card, right? As long as it's not an actual credit card? Also, some banks do the fees the other way around - they charge you for debit transactions but not for credit transactions.