If the phrase was "a cardboard cut out of a dime," it would indeed mean that the cardboard was made by cutting a piece from a dime. Obviously that's impossible, so it's a good thing that's not what it said. A "cardboard cutout of something" is a likeness made from cardboard and cut along the outline of the object.
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I believe only the sense #1 could be considered relevant here but still I don't see how it set in the given setting. Because a cardboard isn't cut out from a dime.
Jackson6612cutout (noun)1 : something cut out or off from something else; also : the space or hole left after cutting2 : one that cuts outThe defiinition is n