Is the anointing of the sick the same as the last rites?

Yes, exactly the same, just a different name; in more modern times the anointing of the sick, and in older times extreme unction. Extreme is used for someone who is at the point of death, and unction is the oil, which has the power either to cure them physically or, more usually, to cure them spiritually. It is in James 5, plain as day: verse 15 says if he has sins they will be forgiven him. Interestingly, verse 14 uses the Greek word astheneo for sick, which usually means physical sickness, but verse 15 uses kamno, which refers to spiritual sickness and is used only three times in the New Testament, each time spiritually. So the Church guarantees the spiritual healing even if it cannot guarantee the physical, and it works ex opere operato, so that even if you cannot verbalize your sins as in confession, the Church takes over and forgives them, preparing you for judgment. There is nothing wrong with receiving the last rites several times, because you do not want to take chances with your own death.