Why? Because it's specifically permitted by Amex that you can use ANY Amex card that's in your name for FHR stays. They give you the perks because you're paying the Platinum/Centuion annual fee. And, as long as you are using an Amex-branded card, they're also getting the processing fee for the transaction.

One reason that I think that Amex offers the "use any Amex card for FHR stays" is the position that several people I know are in: they have a personal Platinum/Centurion card, but often travel for business and have a business-issued Amex for business travel. If they put business travel on the business card, then it gets paid for by the company automatically, without having go through a reimbursement request process. This "any Amex card for FHR stays" feature allows them to use the business card for FHR stays.