Are they unique? Well, yes and no.
Technically, there's only one of each Legendary in their respective games. In that sense they're totally unique. In-game, the only evidence you need (or that there is) is that they're one-of-a-kind and nobody else has caught a Legendary.
But going meta, the fact that everyone is capable of catching each Legendary makes them just as common as Pokemon found in the wild. Say you got together with six friends that had Soulsilver, and you traded with each of them to get the Lugia from each friend. You now have an entire team of Lugia, which would go against the "one-of-a-kind" thing I just mentioned. But as with all Pokemon, each would likely still have a different Nature and/or Characteristic; this technically makes them unique among themselves, even if they're no longer one-of-a-kind.
Short answer, all Pokemon, Legendary or not, are unique. Legendaries are even more unique, and even more so in-story. But on the global spectrum, they're not much different from other Pokemon in that regard.