How can I pay for photoshop CC without using a credit card?

A debit card, if it can be used as a credit card, can be used to pay for anything online. If he's using his debit card to pay for his CC account then it is possible. The rep may have just been referring to using a checking account being a method that isn't possible.

He needs to stop using his debit card to pay for anything online as debit cards are unsecured so if someone were to acquire the card number they could clear out his checking account and your son would be unable to do anything about it.

I am using a Visa-branded local bank debit card for my CC subscription payments.   I don’t think you can enter a routing number and a bank account number for ACH.

A credit card likely provides more protection against fraudulent charges than a debit card, though, so there are benefit for having to pay a little interest every month.