The FAA requirement has been reinstated. At this time there are no known additional restrictions for AMA members.
At AMA chartered sites you are required to be an AMA member to fly and your plane has to be marked with at least your name and AMA number per the AMA safety code. The FAA requires your FAA registration number also be on your plane and the number has to be located in a place that does not require tools to access them.
AMA Safety Code