What happens to Attachments (say Milk of the Poppy) when a duplicate character is used to save the character, does it remain?

Attachments remain because:

When a character is saved (with a duplicate or any other card effect), that card never dies . Since it never dies, it never leaves play. Since it never leaves play, none of the things that happen when a character leaves play (returning non-Terminal attachments to hand, discarding power, ending lasting effects, etc.) apply.

It's a common misconception that saves work by "resurrecting" the character, returning it from the dead/discard pile after it dies. But instead, the kill/discard effect fails for that character and it simply never leaves play.