Pinocchio is born dishonest. This is both true in a symbolic way, as well as a literal way. Quite literally the first thing Pinocchio does after being born is get Geppetto thrown in jail:
"Poor Marionette [Pinocchio]," called out a man. "I am not surprised he doesn't want to go home. Geppetto, no doubt, will beat him unmercifully, he is so mean and cruel!"
"Geppetto looks like a good man," added another, "but with boys he's a real tyrant. If we leave that poor Marionette in his hands he may tear him to pieces!"
They said so much that, finally, the Carabineer ended matters by setting Pinocchio at liberty and dragging Geppetto to prison.
Pinocchio's self-portrayal to the world is already dishonest, and his inability to claim ownership over the bad he does is shown to be the central problem. And the story has barely even started. Pinocchio is born dishonest, a liar in a literal sense.
Thankfully, Collodi does a great job of pointing this out to us at every turn:
The story of Pinocchio and the Talking Cricket, in which one sees that bad children do not like to be corrected by those who know more than they do.
It's pretty obvious from the chapter headings - not just this one, but others included - that Pinocchio's portrayed as a child who was born "bad." The nose plays into this. Heck, minutes, maybe hours into his existence, he's already committed his first murder:
Perhaps he did not think he would strike it. But, sad to relate, my dear children, he did hit the Cricket, straight on its head.
With a last weak "cri-cri-cri" the poor Cricket fell from the wall, dead!
Lying is ultimately used as an analogy for all of Pinocchio's bad behavior. Many of the subsequent sections of the book involve Pinocchio doing something bad, and then lying about it. Collodi helpfully lays the first instance out for us as a chapter heading:
Pinocchio eats sugar, but refuses to take medicine. When the undertakers come for him, he drinks the medicine and feels better. Afterwards he tells a lie and, in punishment, his nose grows longer and longer.
Ultimately, the meaning of his nose growing is to set a precedent. When we later encounter the Curse of the Growing Nose, we know that it's because something about the way Pinocchio was made... makes him lie. He was born bad, and that's the point.