If we are to serve God alone, why are servants told to obey their earthly masters?

The words are different. In Matthew 4:10 ‘serve’ translates latreuō, which the New Testament uses only of religious service and worship; Jesus is answering Satan’s demand to be worshipped, and worship belongs to God alone. Matthew 23:10 is about religious teachers: kathēgētēs means a teacher or guide, and Jesus warns his disciples not to set themselves up as masters over one another’s faith, as the scribes did.

Paul uses the ordinary word for a slave’s work, douleuō, and applies it to the Lord: in doing their earthly work well, servants are serving Christ. They are to work “as to the Lord, and not unto men”, because “ye serve the Lord Christ” (Colossians 3:23–24). That obedience has limits: it covers the work, not the conscience, for “We ought to obey God rather than men” (Acts 5:29), and Paul reminds masters that they “also have a Master in heaven” (Colossians 4:1). Serving God is the reason for faithful service to people, not a rival to it.

Slavery

Some critics read these passages as approving slavery. They give rules for Christians living under Roman law; they do not call slavery good. Paul urges a slave who can gain his freedom to take it (1 Corinthians 7:21, as most modern versions read it), and tells believers, bought with a price, not to become “the servants of men” (1 Corinthians 7:23).