I think it depends in the game and the goals and expectations for it.
So, I’m currently playing in two games. In one, I’d say character is not very important at all. It mostly amounts to choices of class options to help the team, a bit of flavor while portraying the character, and perhaps a sliver of motivation in certain instances. The game is very much story first. We’re playing D&D and going through the classic module Temple of Elemental Evil.
My other game is probably more about character in that if we changed the characters, the events of play would go differently. The story that emerges from play is the story of our characters specifically. They cannot be swapped out for others and have the game go the same. This game is Stonetop.
What makes these two games different is largely the rules and processes of play. Those rules highlight story or character in both active and passive ways.