Shadow's game also mischaracterized him as an uncaring jerk. Even in the opening cut scene before the first level begins, he sees the Black Arms aliens coming down from the sky and beginning their attack on the city, harming and terrifying innocent civilians, and he's just like "Whatever, not my problem." He only gets himself involved after Black Doom appears before him as an apparition and vaguely intrigues/puzzles Shadow enough into wanting to collect the Chaos Emeralds.
But regardless of whatever damage this game did to Shadow's characterization, it was very short-lived since Sonic '06 went and immediately course corrected by writing Shadow in a way that was much truer to his depiction at the end of SA2.
The actual most damage done to Shadow's personality was in the Boom games in cartoon, in which shadow was basically depicted as a parody of early-DBZ Vegeta, constantly wanting to one-up Sonic and gleefully insulting him, his friends, and their friendship, like an insecure schoolyard bully who's trying way too hard to sound tough, macho, and cool. And while the Boom cartoon was mostly a parody show in and of itself, both it and the Boom games played this try-hard version of Shadow completely straight, with him genuinely believing that friendship and compassion are weaknesses to be mocked and spat upon.