Why do we think that Jesus is the only Savior?
Because of all the people who have lived and ever will live, Jesus alone qualifies, in his person and work, as the only one capable of accomplishing atonement for the sin of the world. Consider the following ways in which Jesus alone qualifies as the exclusive Savior.
Christ alone was conceived by the Holy Spirit and born of a virgin (Isaiah 7:14, Matthew 1:18-25; Luke 1:26-38). This alone qualifies him to be the Savior. Why does this matter? Only as the Holy Spirit takes the place of the human father in Jesus’ conception can it be true that the one conceived is both fully God and fully man. Christ must be both God and man to atone for sin (see below), but for this to occur, he must be conceived by the Holy Spirit and born of a human virgin. No one else in the history of the world is conceived by the Spirit and born of a virgin mother. Therefore, Jesus alone qualifies to be Savior.
Christ alone is God incarnate (John 1:1-18, Hebrews 1:1-3, 2:14-18, Philippians 2:5-11, 1 Timothy 2:5-6). So, Jesus alone qualifies him to be the Savior. As Anselm argued in the 11th century, our Savior must be fully man in order to take the place of men and die in their stead, and he must be fully God in order for the value of his sacrificial payment to satisfy the demands of our infinitely holy God. Man he must be, but a mere man simply could not make this infinite payment for sin. But no one else in the history of the world is both fully God and fully man. Therefore, Jesus alone qualifies to be Savior.
Christ alone lived a sinless life (2 Corinthians 5:21, Hebrews 4:15, 7:23-28, 9:13-14, 1 Peter 2:21-24). Because of this, Jesus alone qualifies to be the Savior. As Leviticus makes clear, animals offered as sacrifices for sin must be without blemish. The animals prefaced the sacrifice of Christ who, as sinless, was able to die for the sins of others and not for himself. But no one else in the history of the world has lived a totally sinless life. Therefore, Jesus alone qualifies to be Savior.