If you attack with Reek and steal a Red Viper. If your opponent draws another Red Viper he cannot play it because one of his copies is already in play.
But that's not the same thing at all.
As Underworld points out, the rules do not say "there cannot be two copies of a unique card owned by the same person in play at the same time." They say "you cannot play or put into play a unique card if there is another copy that you own or control already in play." It matters a lot that the rule talks about what players can do, instead of talking about an "illegal game state" as a result of some action.
So, in the original example, when you pull Widow's Wail you say "I am trying to put into play a copy of a unique attachment; do I own or control a copy of this attachment already? No? Is there a copy of it in my dead pile? No? Then there is nothing preventing me from putting the card into play." What your opponent owns, controls, or has in his dead pile never comes into it.
This is similar to the reasoning that if you have a copy of a unique character in play, then someone puts a copy of that same unique into your dead pile (say with Aegon's Hill), nothing happens to the copy that is already in play. It's not having a unique character in play and in your dead pile at the same time that is illegal; it is the action of playing/putting the unique character into play while a copy is in your dead pile that is illegal.