Ok, so if I'm understanding how the unique rule works, you can only have one unique card in play with that title at a time. So, how does the unique rule apply in the situation of ShaggyDog the character and Shaggydog the attachment?

It works exactly the way you just said. You can only have 1 copy of a unique card, by title, in play at a time. Card type doesn't matter. If you have the attachment version in play, you cannot play the character version, and vice versa. If you have the character version in your dead pile, you cannot play the attachment version, either. On the other hand, you can dupe the character with the attachment, and vice versa.

Another important point, you can only have 3 copies of a card, by title, in your deck to begin with. So you can only have a total of 3 cards named "Shaggydog," regardless of type, in the first place.