Why does a sponsor have to be a baptized Catholic?

Yes, the sponsor for baptism or confirmation must be Catholic. See canons 874 and 892-893 in the Code of Canon Law. It's the sponsor's job to help bring the person up in the Catholic faith and to " take care that the confirmed person behaves as a true witness of Christ and faithfully fulfills the obligations inherent in this sacrament." See the full requirements below. (Canon 893 specifies that the same requirements listed for a baptism sponsor also apply to a confirmation sponsor.)

1/ be designated by the one to be baptized, by the parents or the person who takes their place, or in their absence by the pastor or minister and have the aptitude and intention of fulfilling this function;

2/ have completed the sixteenth year of age, unless the diocesan bishop has established another age, or the pastor or minister has granted an exception for a just cause;

3/ be a Catholic who has been confirmed and has already received the most holy sacrament of the Eucharist and who leads a life of faith in keeping with the function to be taken on;

4/ not be bound by any canonical penalty legitimately imposed or declared;

5/ not be the father or mother of the one to be baptized.