Some would believe that the there is one Blue who appears in both the manga and the games; however, I find that they are in fact two separate beings. Based on their personalities it is hard to believe that they could be the same person. The Blue from the games is impulsive, rude, and jealous, while the Blue of the manga is cold, strategic, friendly, and confident.
There are a few similarities between the two Blues. Both lose to Red in a battle to become the Pokémon League Champions. Later in their respective series both become the leader of the Viridian City Gym. Both are confident in their abilities and believe in their skills. When compared to their differences, however, their similarities are few.
In the Pokémon games Blue plays the role of rival. He is constantly picking on Red because he believes it will make his grandfather love him. He treats his Pokémon like pawns and doesn’t form a bond with them. In the manga Blue and his grandfather have a close relationship and Blue will risk his life to help him. Blue and Red are both friends and rivals who respect each other. This Blue treats his Pokémon as partners and family. He respects them and won’t just use them.
Overall, the differences far outweigh the similarities between the two. With their completely different personalities and relationships it is hard to believe that they could be one person. They are more likely two separate entities from different forms of media.