I like to think of it in this way: In Pokemon there are three main "universes": Games, Manga and Anime (well, four including the manga based off the anime, but it has too many similarities to be called its own universe).
In the games, the main characters can have any names, but are officially caleld Red and Blue/Green (English/Japanese respectively). Red can do whatever he wants, catch whatever Pokemon he wants, etc, because he is controlled by you. In GSC it is supposedly the same Red, but in the future, with a team of Pikachu, Espeon, Snorlax, etc. In the games, Blue/Green is Red's rival and the Elite 4 Champion in RBY, and a Gym Leader is GSC.
The manga is pretty similar to the games, except Red and Blue/Green are slightly different to their game counterparts, with slight differences in appearance, personality, journeys and what Pokemon they catch. Since I do not read the manga I can't really go in-depth with this one.
In the Anime we don't have Red and Blue/Green, just Ash and Gary. Ash, unlike Red, starts off with a Pikachu, and catches his own set of Pokemon. He also travels around with Gym Leaders Misty and Brock (and later, May and Max), which Red did not do. Gary is pretty much the same personality-wise as Blue/Green, but he does not do as well as a trainer and decides to become a professor (which you already knew).
Basically, in every universe the characters are different, despite all being based off the same concept, and sometimes even having the same name. But, think about it, wouldn't it be boring if the games, manga and anime were exactly the same? Most likely...
