If you peak through the window of an elementary school classroom, all the kids are being taught by the same teacher, they are all receiving the same instructions, but if you look at their drawings displayed on the bulletin board, each child has a distinct style from the very beginning. Each child's work has certain identifying elements that are always there, in every drawing, in every painting, that are as unique and recognizable as a fingerprint. The colors a child uses follow a pattern, the way they lay-out their drawing follows a consistent set of rules, maybe only consciously known only to some deep part of that child's brain, but recognizable to our pattern-seeking eyes as unique rules, even if we can't quite put our finger on what those rules are...
But are the drawings good...
Taking that style, and adding technical ability over the years is the key. Mastering the techniques that will allow each person's particular style to be used in the most expressive way possible, so that everything supports that style, rather than detracts from it, and so that that style can be used in the service of all the things that one person needs to say, that is the difficult part. One must boldly set aside the fear of being too changed, and embrace every tool that may be of help. Rush at education with open arms, trusting that the good and valid parts of you will always be there, impervious to new thoughts and new habits, and that only the chaff will burn away, leaving you, ultimately, a purer and truer version of yourself.
So, in short, I think it's some of both... innate identity, there from the very beginning, encoded in our genes, and the conscious polishing of that identity that makes it something of value to other people. Some people die before they can get polished (or published.) Some people don't have something to say that speaks to their particular time. Some people get stuck with the wrong teachers. But for the lucky few, the stars align and we all go "Ooooooh!"