This entire thread seems somewhat missing without the famous predecessor article Against Narrativity
http://lchc.ucsd.edu/mca/Paper/against_narrativity.pdf by the same author.
The linked article provides some phrases/concepts which seem more amenable to grappling with/interacting with. Specifically, it breaks the concept of Narrativity into the psychological Narrativity thesis (That it is human nature to experience their lives as a story or collection of stories, with narrative arcs and plot points as we're used to describing in fiction),
and the ethical Narrativity thesis (That the act of experiencing/conceiving of one's life as a narrative is a capital-g Good Thing, and that narrativity is essential to full personhood).
From there, it splits outlooks based on truth values. One could think that humans are innately narrative-driven in their thoughts and that this is good (many people here). Alternately, one could think that people are narrative-driven and this isn't good (This also shows up in the article). Contrarily, one could think that people aren't innately narrative (but that it's good to aspire to), or lastly that narrativism isn't innate, and that it isn't good to assume that this is the case.
I find the last one compelling, myself. Many people do continually tell themselves the story of themselves, but this isn't innate/required. One can have a series of events, see the causality, and have plans for the future without a Narrative. Believing in the Narrative would suggest things like believing that your life is part of the Campbellian Hero's Journey.
This is sometimes helpful, as it can suggest to people that they can hold on for things to get better, but it's also where we get things like the decried Nice Guy behavior where men expect that inputting kindness coins leads to the sex payoff, because those are the stories they're immersed in and the ones they believe in. I've seen schizophrenics describe their mental models as taking this pattern-matching wiring in humans and having it turned up to eleven. Seeing data points and watching connections form to where there's a Grand Narrative connecting things. (Not a point against them, though to me at least it does point towards narrative pattern-matching as something people can vary more and less in feeling tendencies towards.)
As for myself, I sometimes tell narrative tales of myself, but it feels like a deliberate thing. I can describe my past in narrative arcs (and it's a really good point made above that there's many incentives to do so in the workplace), but it's not something I feel is actually the case or needs to be the case. "Plucky young farmboy gets into computers, is drawn to the Big City by Megacorp, has a fall-from-grace there". Where does the story go from here? Under a Narrativist mindset, there's some predictive capacity here. Fall-from-grace leads to soul-searching and redemption? If we could use narratives to predict the future, that would definitely point towards Narrative as a driving force. On the other hand, being able to say "No, my story doesn't have to go like this, I don't have to have a story leading me in this direction" is powerful. By throwing off those chains of narrativity, it allows some measure of greater control.
So yes, I think people are drawn towards personal narratives. They're compelling. We're surrounded by stories all the time. There's incentives to talk about one's past in that way. But they can be confining, and they can lead to harmful behavior as people are immersed in stories suggesting specific arcs.