I think that the "Rule of Six" relates to the old Transmedia card, which was the progenitor of the current iDine and airline dining programs. Transmedia was a separate card that was linked to the customer's credit card, and Transmedia refunded a portion of the meal cost to that credit card after the whole transaction was processed.

Because the restaurants bore the cost of this refund (usually indirectly, through private business loans), the restaurants would only accept the Transmedia card for parties of six or fewer (unless a larger number was approved by the restaurant in advance).

While the Transmedia card existed as a separate card, this was easily enforced. Now that the transaction is invisible to the restaurant, the enforcement of this rule is more difficult.