To be fair, every game past the 1st generation never had any indication that what the player character did ACTUALLY happen, or if anything...just allude to storyline in passing.
Like Cynthia mentioning in BW that the PC's eyes reminds her of another trainer's when they fought against Giratina, though she didn't mention a gender or anything.
Red gets by because he was simply the first trainer, his fame has gotten to such an extent that his team changed between GS and HGSS because....they didn't want people confusing him with Ash from the anime, nor with the Red from the Manga since His Eevee evolved into an Espeon since the original release and didn't want people confusing him. replacing one of his Pokemon with a Lapras.
That and G/S was originally going to be actual sequels to R/B, they probably dropped off the numbers during late development when they found out how much of a cashcow it was around the world. --- RSN: Plucky9, Total: 2340 KoL: Plucky, currently a Accordion Thief