Literally in the first Pokemon game on the good old Gameboy there is the system where the Pokemon are send to and stored on "someone's pc". There is no post box system but just a PC where you can send (and retrieve) your Pokemon from anywhere on the continent in a split second, a speed not possible with the postal system.

Starting in Let's Go you can access the Box system from anywhere - and it's called the Box Link. Unless the trainers all secretly cart off their pokemon to a mailbox while we aren't looking with the menu in our way it's probably digitizing.

But where are the pokemons that you catch when your team is full?

It’s a save editor. You can modify or create new Pokémon with it