As others have mentioned, it's not possible to register the same credit card to different dining programs. Though, it's possible to double-dip and accrue miles in different programs if you register a credit card from one airline to a dining program from a different airline.

For example, if you have a Delta Skymiles Amex Card registered in the AA Dining program. If you use this card when you eat out you will earn miles for the transaction in Skymiles, while the miles you earn for dining at that restaurant would go to AA.