I agree that this is not her signature. Nor is it, I feel, a fraudulent item, just someone labelling a picture in a collection or perhaps a newspaper person, any markings on the back? Some people have poor facial recognition and all pretty blondes look the same :-)
I have not seen her reduce her name to MM, but just M, as in this postcard to close friend Ralph.
It's really too little to go on as well. A hopeless case and not worth buying.