In the game:
Yes, they are you unique. There is only one of each legend available in the game (Ex: HG/SS; only 1 Lugia or 1 Ho-Oh). The only legend that you can get more of is Phione, but:
Phione's legendary status is disputed.
Phiones born through breeding cannot evolve, so no real legendary Manaphies.
So yes or no, depending if you see Phione as a legendary.
In the Anime:
Not all Legends are unique as you put it. It has been pointed out that Lugia has a baby named Silver. Also, from Bulbapedia:
... However, like all creatures, they can become enraged if they feel threatened, particularly if they are protecting their young.
I'm sure there are other instances of legends breeding.
... There are some cases where legendary Pokémon are implied to breed in the wild, as seen with Lugia, though this is not the case in the games.