Unique cards are algorithmically generated, which means their stats, mana costs and other text rules or faction (one of the two factions from the Rare versions) vary from one to another.
There can be multiple copies of a unique card from the same family, but all with a different generation of stats and rules.
All unique cards are purely unique. The generated one you own can never be found elsewhere in the exact same combination of family and stats/rules etc.
This is also true for uniques being two distinct cards with different card names.