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I think the mint is the analogue store.
There is a lot of language in the gift card laws about the card only being redeemed at the company it came from. This is true of fedimints and Cashu mints I think even in the case where you are interacting with a LN gateway or another mint. The gateway and mint both have to "redeem" the ecash at the original mint. So that's another similarity.
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Sure, but who's the analogue of the store, then?
Is it more like a Visa gift card? People probably think of that more as Visa having custody of your money.