It is known that there is no reality without someone who senses the reality. Hence, when we say “Nefesh de Atzilut,” it means that we are sensing a certain measure of attainment in the Upper Abundance, a measure which we call Nefesh.
And world refers to the “common” within that attainment, meaning that all the souls have a common form so anyone who attains that degree attains that name, Nefesh. This means that it is not necessarily that a specific individual attains that name and in that form, but that anyone who achieves that degree—which is certainly through the preparation of Kedusha and purity—the abundance appears to him in that form, called Nefesh.
We can understand that from a corporeal example, applied in this world. For example, when one says to another, “Now I am going to Jerusalem,” when he says the name of the city, everyone knows and recognizes that city. They are all certain of the place he is speaking of, since those who have already been to that city know what this is about.