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Accurate statement or not, how many people in 1938 had seen a “real card” with which to compare? Having been born several decades later, I think I’ve seen one real SS card in my life: my own. And I lost that years ago.

Maybe they mean that the card was physically the same size, but the printing was scaled down to only fill half?

Maybe the real cards were printed on one side of paper and you folded it in half? I remember having a card like that, though it may have been for the Selective Service. Otherwise I would guess it's just a mistake.

Half the thickness?