Apart from the ones already mentioned, a few more 
Submitted but not released (demo is still up if you’re interested though) The Saga of Oedipus Rex where you play as Oedipus from the Greek tregedy.
Most of @Mayday’s games have relatively preset protagonists like Twin Flames
Divided we fall follows the story of 4 specific people in the Spanish Civil War.
Not sure if this is quite what you’re looking for, but in 3Games you control various members of the party from chapter to chapter in a hunger games like story.
Edit: WIP shout out to Donor where you’re a girl with anxiety issues that’s been trapped by two vampires. (Some cutomisation, but other things like the xanax habit and how that affects what happens is mandatory to at least a degree.)