If a card is captured, it is supposed to lose all its characteristics and simply become a "captured card". So, if a unique card is captured, can copy of that unique card be put into play before it is rescued?

Of course. A captured card has no title, and uniqueness is checked by title, so you can play a new copy.

Radiskull is correct, but I want to add something.

When you capture a card it is placed face down, And per the rules, any face down card is considered out of play.

Some card effects instruct the DS player to capture a card. Only LS cards can be captured. This is done by attaching the captured card facedown to the DS objective designated in the effect’s text. If no DS objective is specified, the DS player may choose which of his objectives the captured card is attached to.

When a card is captured, discard all its tokens, enhancements, and return its Force card (if any) to its owner’s supply. A captured card has no characteristics (i.e. no text, type, numbers, or other information), and is considered solely a captured card until rescued or discarded.