Actually only a hand full of Legendary Pokémon are consider Unique (One of a Kind).
Pokémon that are not considered unique are:
Mew (Mewtwo Strikes Back and Lucario and the Mystery of Mew)
Mewtwo (Mewtwo Strikes Back and Genesect and the Legend Awakened)
Lugia (The Power of One and a young Lugia called Silver EP220, EP221, EP222)
Celebi (Pokémon 4Ever and Zoroark: Master of Illusions)
Regirock, Regice, Registeel (Brandon's Regis and Lucario and the Mystery of Mew)
Latias (Pokémon Heroes)
Latios (Pokemon Heroes and Tobias’s Latios)
Jirachi (Jirachi: Wish Maker and BW132)
Deoxys (Destiny Deoxys)
Heatran (Arceus and the Jewel of Life and DP169)
Regigigas (Giratina and the Sky Warrior and DP129)
Manaphy (Pokémon Ranger and the Temple of the Sea)
Darkrai (The Rise of Darkrai and DP104)
Shaymin (Giratina and the Sky Warrior)
Genesect (Genesect and the Legend Awakened)
Pokémon that are debatable and could end up in either category:
Articuno, Zapdos and Moltres- We aren't actually aware if there're unique or not, they have appeared in several different episodes and in The Power of One but they do seem to travel quite often so could be the same. Since Nolan did not officially catch the Articuno he used against Ash as they were just friends we cannot place the Legendary Birds in either section so I deem them in the Debatable Category.
Entei, Suicune and Raikou- Although we are aware that Ho-Oh created them from the ashes of Pokémon, Suicune has appeared multiple times and even in different regions. It has appeared in three episodes of the anime and Pokémon 4Ever. Entei has also appeared in several episodes and in Spell of the Unown. Although the Entei in Spell of the Unown was only an illusion created by the Unown. All of the Legendary Beasts also appeared in Zoroark: Master of Illusions where they are shiny. Even though they are normal in the Anime and shiny in Zoroark: Master of Illusions they are still placed in the Debatable Category as Ho-Oh could have potentially changed their forms as it is the individual that created them.
Cresselia- One question if there's more than one Darkrai, why not Cresselia?
Victini- Victini is in the Debatable Category because there has only been one seen so far, this is in Movie 14. Even though Victini is probably a unique Pokémon, each of the Legendary Pokémon with a overall stat of 600 are in the non unique section so it get's put in the Debatable Category.
Meloetta- It has only been seen once in the Anime so far.
Diancie- Although Diancie has only appeared once in the Anime, Diancie doesn't seem to be a unique Pokémon, as Diancie is born from a Carbink that undergoes a mutation.
Pokémon that are consider unique are:
Ho-Oh - Even though its counterpart, Lugia, is able to have offspring, Ho-Oh seems to be a single Pokémon that travels around.
Kyogre, Groudon, and Rayquaza - Kyogre and Groudon have appeared in the Team Magma and Team Aqua arc, and Kyogre has appeared in Pokémon Ranger and the Temple of the Sea. Rayquaza could be argued as being more than one of its species, but it is a high probability it is the only one.
Azelf, Uxie, and Mesprit - They were created by Arceus to keep Dialga and Palkia in balance.
Dialga and Palkia - They are the keepers of Time and Space.
Giratina - It is the guardian of the Reverse World, an distorted mirror of the Pokémon World. Although it could be argued that because mirror dimensions exist, that there would be Reverse Worlds for each of these mirror dimensions, that is extremely technical and probably won't be mentioned.
Arceus - Creator of the Pokémon Universe. Not much more to say about that.
Cobalion, Terrakion, Virizion and Keldeo- Only appearance was in Kyurem VS. The Swords of Justice. Seem to be a unique group.
Landorus, Tornadus and Thundurus- Only appearance was in BW059 and BW060 but seem as though they are unique.
Zekrom and Reshiram- Even though Zekrom and Reshiram have made several appearances in the Anime and have both appeared in Movie 14 they are considered to be unique as they represent Ying and Yang in the Pokémon World.
Kyurem- Part of a trio with Zekrom and Reshiram.
Xerneas and Yveltal- They both made an appearance in Diancie and the Cocoon of Destruction, Xerneas also appeared in XY041 although as silhouette, so we can not determine if it was a different Xerneas or not. However Xerneas and Yveltal are considered unique as they represent Life and Death respectively in the Pokémon World.
Zygarde- Zygarde has been seen at all so far in the Anime, but it is assumed that it is the the only one of it's kind.