Are you sure this is how it works? In my experience, it is the other way around: when logged/signed in my AmEx Membership Rewards, I just need to add my FB number (or any other airline FFP / hotel loyalty program). I would need to confirm my identity by providing the CVV code and/or the 3 digit code on the back of the card, and the linkage is done instantly.

I am not aware of any location in the AF/KLM/FB website where you link your AmEx information nor any other partner (to the exception of maybe the Accor "Points+Miles"). FB is merely a recipient of any transfer, so it makes sense that the linkage process takes place on the place that transfers out the points.