One system that I play and GM - Burning Wheel - charges PC-build resource points for relationships. These are cheaper if they are close family (and hence, everything else being equal, looming larger in the life of the PC) and cheaper if they are enemies rather than friends (and hence, everything else being equal, hostile to rather than supportive of the PC).
One thing the player is paying for when purchasing a relationship is the focus of the fiction: the GM is obliged to incorporate relationships into the game. By spending the additional points required for the PC to be a friend rather than an enemy, the player is paying (i) for the GM not to have the character oppose the PC, and (ii) for the prospect that adversity in the game (as established by the GM) will include his/her PC's commitments to friends and family being called into question.
In its details this is nothing like the system @
Hussar described, but I don't find it odd that a RPG should include rules that allow a player to direct the GM as to how a certain bit of background may be used. I certainly think the suggestion that it's either open-ended for the GM to decide, or else "fan fiction", is wrong.
I have played, mostly ran, many systems where NPCs are "bought" as a part of chargen, much as 5e allows for backstory and backgrounds and classes to also provide openings for player initiated relationships.
I love them.
In many of the "build-all-stuff" systems the values can vary greatly depending on whether they are seen as a boon or a bane on the whole, how useful, how often etc. In others, they are recognized as more explicitly "scene time gains" and so there wont usually be "negative points awarded for enemies."
again, all good.
I tended to describe it (as did players to other players) like this: "In this game, we are adventurers and stuff will be ahppening to us and around us. These things are not "opening us up to stuff" because we are not gonna show up and spend the night knitting. What these do is allow us to have some choice and say in the types of things we want to see."
So, for mostly the whole of my gaiming after say the first couple years, this kind of thing has been the norm and i welcome it and encourage it at my game.
What i am objecting to is the opposite - adding stuff to the game that may include even character class elements and then declaring that as off-limits, cannot be used, etc.
At my table, my response would be "no" and if their views held after a discussion it would still be "no" and i would point them to the gazillion other character choices or backgrounds that don't necessitate BOTH adding in relationships *and* freezing them out from the world*.
Some seem to object to that.
Oh well.
But that is *not* at all the same thing, to me the opposite, as allowing players in their characters to help direct the play and the story by adding elements they want - whether there are rules involved, points involved or not.
EDIT TO ADD Many of the systems allowed things to be bought after chargen too, and some might even require it if the relationship was formalized into the gameplay. i tend to more favor the "not points, preferences and discussion" approach to the "buy your NPCs " approach but thats just me. i never saw "but i can buy it" rules as "enabling" so much as "limiting" myself - tho i have sure ran into point-buy-philes who seem convinced if you cannot put it on the sheet with points it cannot exist or matter.