Since there isn't any Pokémon labeled as the "Omega" Pokémon I have a theory.
Arceus is the creator of all Pokémon and is meant to represent God or some sort of religious figure. The word "Alpha" refers to the "beginning", while "Omega" refers to the "end". If we treat Pokémon as a creation of God (Arceus), there must be a counterpart. This counterpart is the Devil, which is widely accepted as Giratina.
Now, let's assume Pokémon die and go into the afterlife (which they do as seen in the battle between you and the ghost of Marowak). Since Giratina is seen as the devil, this could be a Pokémon's end. Therefore, Giratina could be unofficially referred to as the "Omega Pokémon".