Professor Oak is Blue’s grandfather. Their relationship is strained because Professor Oak is often too busy caught up in his work to give Blue the attention he needs and deserves. Blue grew up without parents, and because of his grandfather’s neglect he only had his older sister to guide him. Blue wanted and needed the attention of an adult and to feel like a member of a family when he was growing up.
Oak’s neglect led Blue to act out. Blue would insult his grandfather in the hopes of getting attention even if it was negative attention. Without Blue realizing it his personality began to change until he became rude and selfish because it was the only way his grandfather would even look at him. Oak even goes so far as to forget Blue’s name. He tells Blue that it’s more important for Red to get a Pokémon and begin his journey than it is for Blue. The worst part is that Oak blames Blue for everything that goes wrong.
BLUE! I'm disappointed! I came when I heard you beat the ELITE FOUR! But, when I got here, you had already lost! BLUE! Do you understand why you lost? You have forgotten to treat your POKEMON with trust and love! Without them, you will never become a champ again! - Professor Oak, after beating Blue at the Indigo Plateu
Blue feels no emotional attachment to his Pokémon and only concentrates on the task at hand (completing his Pokédex). Why? Because, his grandfather raised him by ignoring his emotional needs in favor of work. Yet, when Blue loses to Red at the Indigo Plateau Oak tells Blue that it’s because of his treatment of Pokémon that he lost to Red. Oak doesn’t realize he was the one who made Blue think it was alright to act that way.
In the three years that passed between the Red/Blue/Yellow/Green games and the Gold/Silver/Crystal games Blue has left home to become the Viridian City Gym Leader and while he’s still irresponsible (constantly leaving his gym), he’s learned to respect his Pokémon. Blue’s sister still lives in Pallet where Blue often talks to her, and will occasionally visit her. Despite Blue’s attachment to his childhood home, he has little to no contact with his grandfather. By stepping away from his family, Blue was able to grow and find his own way.