That is not what it says. There is two parts to it: his conception and his bringing-forth (shaping/forming).
Of his conception, he says that his mother was in sin when she conceived him. The word used in the Hebrew for conception is also used to describe when the animal is in heat. Furthermore, the pronominal suffix shows us that it is in context of his mother's heat, or more accurately, "the heat of my mother". So he is saying clearly about the moment(s) his mother conceived him, she was doing sin.
The bringing-forth, or shaping, is telling of the struggle to be delivered into the state of being (
https://biblehub.com/hebrew/2342.htm) and as such it does describe a long period of time, whether in context of being born through a womb or the childhood (which is more likely given that it is said to have been wrought in iniquity).
Therefore he says that the sin and iniquity are associated with the two distinct aspects of his being given the life he has: where his conception was done in sin and his bringing-forth was wrought in iniquity. I think if I was to write it in my own way in English, I would say that my mother conceived of me sinfully and I was shaped by injustice.
It therefore shows clearly that he appeals to God for mercy because of the harm done to him that has put him in a position of undeserved corruption, and thus he believes that his sin is the result of effectively having been robbed of his intrinsic nature. I think that the plea for mercy on those grounds is hands-down the most reasonable given that it is obvious in the facts that he is remorseful.