No, in Red/Blue your character is based on Ash. When you start the game you can choose the name Ash or your game colour (Red on Red, Blue on Blue). Then your rival is either Gary or the opposite colour. It's the same in both games.
I checked my games and in Ruby you choose between Landon, Terry, Seth or Tom. You don't choose the rival's name. In FireRed you must input your own name. For the rival the choices are Green, Gary, Kaz and Toru.
So I think you're wrong. The choice in the early games was your own game colour and the opposite colour for your rival. Nowadays you don't get extra choices for your own name and just type your own name.
And just to make clear: the anime is SEPARATE from the games, the games are SEPARATE from the manga, and the mange is SEPARATE from the anime. Don't try to compare them. Case closed.