There is a calculation that can be made if an AAA number is valid, but that doesn't mean its actually a valid membership. Similar to how you can generate 'fake' credit card numbers that match their checksum calculation but aren't actually valid (closed account, etc). I am guessing that Marriott is doing the former, so you just need a number that matches the checksum.
I made a few reservations this weekend and was prompted the first time when I used the AAA rate but not after that -- but that was all one session so maybe it just remembered for that session. I too have mixed feelings about requiring the AAA number, especially for those that are in partner organizations that should have access to AAA rates but not an AAA number. But if they are going to do it, should definitely store it in the profile. Note I DO have a valid AAA membership and actually use it all the time at Marriott -- at least for me I get rates just as good as Marriott Rewards but often without the prepay requirements. So for me its been worth it to have more flexible travel plans.