made by Neil Druckmann, who was also the writer for the game.

That's an important key. He didn't buy the rights to someone else's work. He basically did a revision of his own work, with all the intimate knowledge of character, narrative, etc. that implies.
The Last of Us as a game was also basically a playable movie as it was. Someone actually made it into a kind of movie by capturing game recordings long before it ever was made as a live action series. That means he didn't have to revise it very far.