The difference between being and doing is often lost in videogames. We have very little idea what or who our character really is, because the character is defined almost entirely by what he or she does. Whether that's saving kittens or punching reporters in the face.
Games as an art form use words and actions as a proxy to express who the character is as an individual. Often poorly, I might add. A character can be just as defined by what they do _not_ do.
I honestly can't think of one that does it well. Games have tried, sure. I think of Tomb Raider letting you explore Lara's home.
The trouble, I think, is game devs having this "projection" obsession. They try to create universally relatable characters. But a character everyone can relate to tends to become a shallow stereotype.
Character is a slippery thing. You or I, we are still the same person before and after a change in opinion. But game characters are often defined by a single attribute; a like or dislike, a political opinion.
I think perhaps Metal Gear Solid: Peace Walker did a good job by having, literally, recorded audio interviews with Naked Snake. Naked Snake is a deep, poignant man. He is not defined by the running joke that he repeats nouns as questions.