From Bulbapedia, the community-driven Pokémon encyclopedia.
A Pokémon Trainer's license, also known as a Pokémon handler's license or simply a Pokémon license, gives the bearer the right to own Pokémon.
Applicants must be 10 years of age or older.
The game equivalent seems to be the Trainer card.
In the anime
In Pokémon - I Choose You!, Ash was excited that he could get his Pokémon license now that he is ten years old. In The Battling Eevee Brothers, Rainer tells Mikey that he could lose his "Trainer qualification" due to leaving his Eevee alone tied to a tree.
In the anime, many children under the age of ten have been shown to own Pokémon, confusing viewers. It is generally an accepted fandom theory that minors under the age of ten may own Pokémon, but are not allowed to compete in Pokémon League-sanctioned tournaments or challenge Gym Leaders.
In the manga
In the The Electric Tale of Pikachu
According to The Electric Tale of Pikachu manga, potential Trainers are required to attend a half-day training session before qualifying to take a brief test. Those who pass the examination receive the license. Only licensed Trainers are legally allowed to purchase Poké Balls.