The word care , as a matter of fact, hardly appears in the Bible in the sense in which we have been using it. I have hardly mentioned God, said nothing about Christ, and not even referred to your salvation. My answer is that it has everything to do with Christianity. Indeed, this is Christianity. Not all of Christianity, to be sure, but it is so large a part of Christianity that 1, for one, can say that it is Christianity.
Christians have no monopoly on care, never forget that.
But Christians believe that God cares about the world, else He would not have made it, and He cares so much about it that one time He became human to save the world from its folly. That human life was the life of Jesus, and Jesus cared. He cared himself, especially for the people nobody else cared for, the people who had been left out in life, and he gave only one test to the people who were to follow him, and the test was this, that they care about their fellow men in trouble. He died because he cared so much, and he lives now in the people who care about him. Christians go so far as to say that God is care, and a Christian might be described as a person committed to the importance of caring.