In the original Pokémon games, Mew was said to have been discovered on July 5. The date appears in research journal entries found in the Pokémon Mansion on Cinnabar Island.
The same fictional records state that Mew was discovered deep in the jungles of Guyana, in South America. Another entry says Mew was named on July 10, while a later entry gives February 6 as the date Mewtwo was born. These dates form part of the mysterious backstory revealed through scattered documents rather than a direct cutscene.
The year of Mew’s discovery is not specified in the games. That is why July 5, rather than a complete calendar date, is the precise answer. The records are also in-universe documents, so they describe Pokémon mythology rather than a real scientific discovery.
Mew was numbered 151 and was hidden from ordinary gameplay in the first releases. Its secrecy, unusual distribution, and connection to Mewtwo helped make it one of the most famous Mythical Pokémon.