Offtopic: No, not a phoenix, just a fiery bird. A phoenix has many more powers than merely being a flaming bird representing fire. Ho-oh is based on the Chinese phoenix straight up.
Pokemon Lore-
Gen 1:
Three birds embody Fire, Ice, and Lightning, and are tied to Lugia. Articuno, Zapdos, Motres
Mew contains genetic code/DNA of every pokemon, symbolizes the origins of life
Mewtwo is the clone of Mew, designed to be the ultimate perfect weapon by man.
Gen 2:
Three beasts after dying in a fire were brought back to life by Ho-oh, symbolizing Thunder, Volcanoes, Wind. Raikou, Entei, Suicine. Because they cannot control their extreme power, they race throughout the land.
Ho-Oh is based on the Chinese Phoenix Fčnghuáng, and is the Guardian of the Skies. This Pokémon is represented by the sun, the sky, rainbows, the color gold, and awakening. It is said to glow in seven colors and see into the future. After the burning of the Brass Tower, it flies around the world looking for trainers with pure heart.
Lugia is the Guardian of the Sea. It is represented by the moon, the sea, storms, the color silver, and slumber. Based on the Japanese sea-kami Ryūjin. Guardian of the legendary birds.
Celebi is the Guardian of the Forest and can travel through time.
Gen 3:
Three legendary golems that guard the Tree of Beginning under their master Regigigas: Regirock, regice, registeel
Latios/Latias: Twin dragons, one male, one female which guard the Water City of Alto Mare. Keepers of the Soul Dew
The weather-related legendary Pokémon: Kyogre of Rain, Water, and Sea, Groudon of Land, Rayquaza of Air represent the balance of nature and play a major role in the storyline of the Hoenn-based Generation III games. It is said that, at the beginning of time, Groudon raised lands and expanded continents as Kyogre expanded the seas. These Pokémon took to sleep after a cataclysmic battle, soothed by Rayquaza. They are based on the behemoth, leviathan, and ziz of Hebrew legend.
Jirachi, the wish giver
Deoxys, a strange alien pokemon with virus-like characteristics which attacks the planet
Gen 4:
Three fairies born from Arceus, representing aspects of the human mind: Knowledge, Emotion, Will: Uxie, Mespirit, Azelf
Three legendary dragons born from Arceus to create matter, one controlling Time, one for Space/Dimensions. and one for Anti-Matter. Dialga, Palkia, Giratina
Cresselia symbolizes the moon and brings pleasant dreams and heals emotional scars
Darkrai causes nightmares and can induce sleep
Manaphy: referred to as Prince of the Sea
Heatran was born from the flames of a volcano when Dialga, Palkia and their relatives first created the universe.
Shaymin: Gratitude Pokémon that curls up in flower-beds and often goes unnoticed. It has the ability to instantly purify an area of all pollutants and transform it into a flowery field
Arceus: the God creation Pokémon said to have been born from an egg in the nothingness of space. It created matter (time and space) and spirit (knowledge, emotion and will) before falling into eternal slumber