A shiny living dex is having all Pokémon that you can catch in a game being shiny. For instance, in Pokémon Sword and Shield it would mean having all Pokémon in both the main dex and the DLC dex being shiny.
Also, it can also mean every Pokémon there is, owning them all shiny. I've seen a lot of Pokémon YouTubers shiny hunting all the time to complete their shiny living dex.
Yeah..."living dex" means you have all of the evos and pre-evos.
For example...having a shiny Charmander that evolves into a shiny Charmeleon and then Charizard would only count as having a shiny Charizard, because, the only one of the three that you would be left with would be Charizard. - You would need to obtain two more shiny Charmanders...and evolve one of them into a Charmeleon, so you are actually in possession of all three forms. - You're essentially doubling/tripling the amount of shinies you need to collect.