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They're similar, although the source of the metaphor is a little different. "has my name on it" seems like it's derived from a barrista writing the name of the customer on the coffee cup.
@MariosAthanasiou Yes, I'm a native English speaker. But I'm not a linguist or language expert, so I don't know many of the technical terms for language constructions. I'm not sure why your answer was downvoted, but so were other answers.