I've recently started a campaign of Spire: The City Must Fall, and there are a couple of bits that (at least partially) address what you're talking about.
The first is less specifically about mechanics, but is very much about a player's conception of their PC, and it occurs during character creation. Each PC gets a pair of Bonds, which are relationships with other characters. One Bond is with an NPC (more on that below) and the other is with another of the PCs. The specifics of the PC Bond are determined by class, but in each case, the Bond allows one player to determine something about another player's character. A couple of examples right from the book:
- You have a bond with another PC- you know a secret about them. Say who it is, what the secret is, and whether they know you know or not.
- You have a bond with one of the PCs who you recruited to the cause. Say who, and say what it was that tipped them over the edge.
So this game actually has a part of character creation where other players can decide things about your character's backstory, and they are potentialy significant and formative events. Now, this has no mechanical expression in the game in the sense of numbers and modifiers and the like, but it's certainly an example of a game rule that supports character concept not being solely that of the player of that character.
Where there are some more mechanical bits is in the NPC Bonds. Each PC has an NPC Bond, as well. Some person they know and can rely on. The way this works mechanically is that each NPC Bond has an amount of Stress associated with it. When you ask them to do something for you, the Bond will take some Stress, the amount of which is determined by the significance of what's being asked, with that being up to the GM. It varies from 1 Stress to D3, D6, or D8 at the highest. At the end of a Session, the GM will roll a D10, and if they roll under the Stress total for the Bond, then that NPC will suffer Fallout.
Fallout is some kind of complication with the Bond. Maybe they're tired of helping you out and your next request of them will be more difficult; maybe the fact that they help you has been noticed by others; maybe they've been made by the powers that be and they're at risk of being executed (the PCs are members of an outlawed revolutionary organization); maybe they've betrayed you. The severity of this is determined by the Stress level and what makes sense according to the fiction.
Mechanics for relationships or that relate to a player's sense of who their character is are often described in these discussions as being unnecessary. And they actually may be, depending on the goals of play, and what any given participant may consider fun. But what they do....and what those who enjoy them do find fun about them....is that they create risk. Your character may not always turn out the way you want them to.....it will depend on what you do, and how the game goes. That can be exciting. That can be fun.
If it's not your cup of tea, that's understandable....some people want everything about their characters to be entirely up to them without any risk or input from others, and that's fine. I would say that such play is on average not going to be as character focused as a game which does include such rules, but whether that's a good thing or not is a matter of preference.
Would you say that D&D supports excitement or action because of the combat rules and the risk to character well being in that sphere?
