I will point out, however, that it is not always a sin, and that's because it's not always false. When it is not false and, therefore, not sin? When God does it.
It is fascinating when you read through Jesus's words and see His constant use of the personal pronoun, "I", in terms of priority and position. All prior biblical prophets kept their finger pointing at "Him", at God. They were not the point; He was. As the last prophet before Christ, John the baptist, said, "He must increase, but I must decrease." (John 3:30) The purpose of the prophet was to speak the words of God for God's sake. Pointing to self as a prophet could not be a positive. Yet Jesus self-consciously and continually pointed to Himself. Consider the many "I am" statements He made along with their huge implications:
Jesus said to them, "I am the bread of life; whoever comes to Me shall not hunger, and whoever believes in Me shall never thirst." (John 6:35)Lots of "I am" statements there, but look at the outcomes. Because of who He is ("I am"), "Whoever comes to Me shall not hunger, and whoever believes in Me shall never thirst," "Whoever follows Me will not walk in darkness, but will have the light of life," "If anyone enters by Me, he will be saved," He died on our behalf, and "Everyone who lives and believes in Me shall never die." Really big connotations to His "I am" claims.
Again Jesus spoke to them, saying, "I am the light of the world. Whoever follows Me will not walk in darkness, but will have the light of life." (John 8:12)
Jesus again said to them, "Truly, truly, I say to you, I am the door of the sheep ... If anyone enters by Me, he will be saved and will go in and out and find pasture." (John 10:7-9)
"I am the good shepherd. The good shepherd lays down His life for the sheep." (John 10:11)
Jesus said to her, "I am the resurrection and the life. Whoever believes in Me, though he die, yet shall he live, and everyone who lives and believes in Me shall never die. Do you believe this?" (John 11:25-26)
Jesus said to him, "I am the way, and the truth, and the life. No one comes to the Father except through Me." (John 14:6)
Let's not forget the singularly most stunning "I am": "Truly, truly, I say to you, before Abraham was, I am." (John 8:58) This straightforward Jewish claim to deity produced the standard Jewish response to blasphemy: "So they picked up stones to throw at Him ..." (John 8:59)
He often referred to Himself in terms that were ... not correct humanly speaking. He said, "I am the one who bears witness about Myself, and the Father who sent Me bears witness about Me." (John 8:18) Big claim. He said, "No one can come to Me unless the Father who sent Me draws him. And I will raise him up on the last day." (John 6:44) You don't get to claim that you are the required permission to come to the Father if you are a standard human being. He told the Pharisees, "You are from below; I am from above. You are of this world; I am not of this world. I told you that you would die in your sins, for unless you believe that I am He you will die in your sins." (John 8:23-24) Now, there's a study in contrasts. In sharp distinction from every other human messenger, Jesus said, "All things have been handed over to Me by My Father, and no one knows the Son except the Father, and no one knows the Father except the Son and anyone to whom the Son chooses to reveal Him." (Matt 11:27) Jesus claims that we cannot know the Father unless He makes it happen.
Jesus was absolutely self-centered in the sense that He was the center and, therefore, actually correct in being self-centered. He stands out in Scripture as such. Being God and, therefore, the center of all that exists, He alone can rightly make this claim as truth and He alone is morally right in doing so. In fact, if God -- the Father or the Son or the Spirit -- was to fail to be self-centered, He would be a liar and idolater.
It isn't really possible to read the many claims and statements of Christ and not conclude that He believed Himself to be the Son of God, God Incarnate, the "only begotten". He was selfless in His relations with people and, yet, as God, He is self-centered and rightly so. If we miss that, we miss something important, because we are supposed to be centered on Him as well. He is, after all, the center of it all.