This is a really complicated matter, and if you were a fly on a wall in a room full of therapists discussing it, you'd very quickly buzz off to another room. I'm going to try to make it as palatable as possible.
The popular conception of a narcissistic person - highly self-absorbed, oblivious to others, ruthless, image-obsessed - is just one of many flavors. Likewise, the popular idea about how someone becomes narcissistic is not entirely on point. Yes, perhaps helicopter parenting and obsessions with achievement and self-esteem can be part of the picture. But in my experience narcissistic difficulties in adulthood are specifically the product of being raised by one or more parents with their own narcissistic difficulties. (Keep in mind that these difficulties may be subtle to the point of near-invisibility or vivid to the point of obviousness or somewhere in between.)
We become who we are thanks to the vast web of interactions with our caregivers. If our care is imbued with acceptance - celebration, even - of the fact that we are at every step becoming our own person, this is truly wonderful!
Parental narcissism, however, is a common impediment. Parents who experience others more as objects than as persons will be prone to relating to their children in this way. And children who grow up in homes where they are related to more as objects than persons are themselves likely to struggle with the distinction as they grow up and attempt to navigate the adult world.
What's more, they may struggle to escape the dilemma of whether they themselves are a good or bad object: the source of joy or pain for the people in their adult lives. The reality, of course, is that they are not objects but persons. The goal of psychotherapy in such cases is to make this a felt reality.
A couple of caveats. One, not all children of narcissistic parents struggle with narcissism. We are raised by forces beyond our parents, fortunately. Two, while a narcissistic household certainly can be an emotionally or physically abusive place to grow up, this is often not the case. This relates, perhaps, to the culture of self-esteem, which I allude to above. When we attempt to boost our child's sense of self-worth, are we operating on an object that we fear will disappoint us, or are we giving our love to a person we know to be separate from ourselves? That is often the crux of the matter.