My read is that each of them represents a different fallout game. Coop is FNV because he's a cowboy, and because he's travelling across the wasteland for a rare McGuffin. Lucy is FO3 and FO4 because she's a vault dweller hunting her family and their mysteries. I feel like that leaves Max representing either the earlier Fallout games or 76, but I don't know them well enough to know if that fits. (Or maybe he represents that one BoS game)