The bank that has issued your first credit card is an Issuer in the credit card value chain. To accept payment for its credit card bill via another credit card, it would need to be a Merchant in the credit card value chain. AFAIK, a single entity can’t be both Issuer and Merchant in the credit card value chain. Which is why, I think, you can’t pay a credit card bill with another credit card.
That said, if you want to gain the deferred payment benefit for your credit card #1 bill, there’s a way to do that. You can find another credit card (#2) and ask them to take over your outstandings on credit card #1. In the industry, this is called “Balance Transfer”. Of course, fees and charges may apply, so shop around so that you make a net gain – rather than loss – out of the transaction.
Should you be interested in knowing more about the credit card value chain, please see my blog post How To Fight Card Payment Surcharge And Take #CashlessIndia To Next Level.