My impression is that co-branded cards which earn Membership Rewards (Schwab, Morgan Stanley) have the same requirements as the proprietary Platinum, Gold and Green cards.
It's correlated, but I think the distinction is between charge cards (generally the unaffiliated ones) and credit cards (most of the affiliated ones).
BTW, as is pointed out here frequently, AMEX seems to put a lot of weight on an individual's history with AMEX. Beyond the simple numbers/formulas/algorithms, a history of opening and closing cards, requesting retention bonuses, and not using AMEX cards much after the first sign up bonus has been earned would be expected to be held against an applicant. Even a history of doing chargebacks more than average could be taken to be a negative factor.