Actually, this approach is fairly common in Japanese RPGs and other story- and character-driven works. Just off the top of my head, here are a few (fairly recent, as in the past 15 years) examples:
Xenosaga
(Shion is presented as the main protagonist but the overall story is really about KOS-MOS, with the various other characters in the group having their own deep background stories that develop the bonds in the party as you play)
Tales of Vesperia
(Yuri is presented as the main protagonist but the story is really about Princess Estelle, with the other characters having their own events and backgrounds explored as the party and main story develops)
Star Ocean: The Last Hope
(Edge is presented as the main protagonist but he's really just the newly appointed captain after the original captain is injured at the very beginning of the game, and he only acts as the captain/leader due to the support of Reimi - who is first mate - and the rest of the party, while the main story is really about humanity's interactions between themselves and other races in the galaxy)
Even is great classics like the original Phantasy Star and the long-running Ys saga, the main protagonist is not necessarily the focus of the main story, plus the various other characters always have a lot of detailed development in order to create the bonds within the group (and emotional attachment for the player).
As I've said, if I wanted a story- and character-focused game, I'll play a Japanese game, including the Japanese visual novels and adventure games, because that's exactly what those works do. I cannot think of such works from Western developers because, even though I enjoy many Western games as well, the Western focus tends to be on action and combat rather than character and story. Even attempts like Mass Effect fall flat compared to the Japanese works I follow, but that's somewhat expected due to basic limitations such as budget, time and other resources. If you spend a lot of resources on action and combat, the resources for character and story naturally take a hit.