This is a good questions. The answer varies from system to system.
The games that influence how I play - which mark me as a FoRE! - are HeroQuest, The Burning Wheel, The Dying Earth and Maelstrom Storytelling. In each of these, the relationship stuff is constrained as part of the character build rules. It is, in effect, another attribute (as it would also be in points-buy game, perhaps).
I'm guessing that, in Flashing Blades, the relationships and statuses that can be acquired as part of PC build don't typically have a mechanical expression for action resolution purposes - or maybe give bonuses for reaction rolls in certain contexts (a bit like some 2nd ed AD&D kits) but the actual reaction rolls are determined using a separate mechanic.
In the games I mentioned, however, the relationship is itself a skill/attribute - so, for example (to cite an example from one of the HeroQuest rulebooks) a player whose PCs has a horse stolen can use his/her PC's relationship with that horse to make an opposed check to unseat the thief. To generalise - relationships become like another skill or talent to be deployed in resolving actions.
Now having said all that - the game that I principally GM is 4e, and 4e does not quantify relationships in this sort of way. So in my game, it's much more open-ended, and relies on the players (with encouragement from me and mutual cooperation among themselves) to specify their PCs in ways that go beyond the character build mechanics.
In my experience - especially when playing with a group of longtime friends, as I do - this is enough to generate relationships and thematic content that do the sort of encounter-driving play that I enjoy.
But in 4e I think relationships are also important at the mechanical level, but in slightly more subtle ways. First, they open or close options as far as paragon paths, epic destinies and some other flavouored character build options are concerned. And in other cases, certain character build choices (eg divine PC, warlock pact, familiar) also bring a certain relationship into existence within the gameworld. This means that there is an ongoing feedback between the open-ended relationship stuff and the mechanical stuff going on - which also interacts with the "developed in play" relationships as well.
The second way that relationships are mechanically important is in resolving non-combat actions, which in 4e are quite abstracted (except for most tactical movement and perception skill checks, and some knowledge checks, which are closer to 3E or Rolemaster in the way they are handled). The GM has to set a DC based on guidelines in the DMG (the notorious page 42!). And at least when I do this, I rely heavily on the relationships the PC has, and the way the player presents their PC's action and the way the context of action implicates those relationships, to help set DCs. So this is not quite the same as "relationship as attribute", but it comes closer to it than a game like Rolemaster, because the use of most skills can often become "expressive" of a relationship, at least to a degree. (This also goes to the point, in the quote in Lanefan's post that is two or three upthread, about "thematic" logic trumping ingame causal logic - hence Gygaxian skillful play has less work to do in my game than in an AD&D or Classic Traveller game.)
I don't know if the above makes sense.
The other thing about these threads is that it's hard always to get a sense of how others play. What I've written above, for example, might make it seem like my approach to GMing 4e is very self-conscious and theory-laden, but in practice it doesn't feel that way most of the time. I also wonder, in practice, what you or anyone else would make of the game I GM. To me it seems fairly vanilla fantasy RPGing with a reasonably heavy emphasis on PC development and an integration of those PCs into a rich mythic history. But then when I see people posting about their strong sandbox preferences, or their strong adventure path preferences, or how they love to play in a really detailed world like the Realms, I think maybe it is at least a little different from how some other people are playing the game! I certainly know that I've played in RPG games that I didn't really enjoy (but that others seemed to) so I've got no doubt that there would be players who'd think my game sucks.
Is Game X a sandbox, linear, matrix? Is it "situation driven"? Railroady? And how do I explain to a player coming into linear-y high-fantasy Campaign X that it's completely different from sandboxy low-fantasy Campaign Y? What is high fantasy, anyway? 
