Originally Posted by sdsearch

AFAIK no one will get a free checked bag (if you're not traveling on an international itinerary where everyone gets a free checked bag).

The free checked bag is associated with the account that has the card. Since there's nothing on this ticket that ties either your wife or you to your wife's account, there's no way to trigger the free checked bags. Your wife could try flashing her card at check-in, but if they compare names on the card to the name on the booking (or if the check-in person is not familiar with the card's benefits), that may not help.

You could go to the AA forum and ask if there is some way to change the FFN on one or both tickets to your wife's FFN. I'm not sure if that's possible at all, and there might be dangers (loss of ability to make changes to flight?) in doing that even if it possible.