This is the covenant aspect of baptism. The one who is baptized received a benefit of atonement through the ordinance of baptism, but in receiving the benefit, a covenant is created. That covenant is to take the name of Christ, and to endure to the end (see the comments following 3 Nephi 12:48 for more information on what it means to endure to the end.)
For the ancient world, taking the name of Christ would have had significant implications that modern man has forgotten. The implication is not simply an identification, but a imitation. We are not just to adopt the name Christian, but the attitude and actions of Christian.