No, you will dispute the duplicate charge using the online facility which anticipates this common occurrence, and American Express will reimburse the other.
It's premature to say if this is an American Airlines problem or an American Express problem.
I would not be surprised to find out something is up with AA's billing mechanism. My Barclaycard got charged for two $303 AA ticket bookings. They were legitimate charges by AA, but the problem is, I never booked AA tix. I had to submit a dispute with Barclays and AA eventually retracted the charges (the whole affair took about 3 weeks to sort out). Wondering if this is some new bug in their billing system.