What does it mean to honor your father and mother in our world? Is an unmarried woman still under the leadership of her father until she is married, or is that different today?
Exodus 20:12
Honor your father and your mother, that your days may be long in the land that the Lord your God is giving you.
Does this apply to men and women differently? No.
Is it actually Biblical that women are under the leadership and authority of their fathers until they are married? Yes.
The Bible tells us how we as children (both boys and girls) are to respond to our parents.
1 Corinthians 11:3
But I want you to realize that the head of every man is Christ, and the head of the woman is man, and the head of Christ is God.
Ephesians 5:22-27
Wives, submit yourselves to your own husbands as you do to the Lord. For the husband is the head of the wife as Christ is the head of the church, His body, of which He is the Savior. Now as the church submits to Christ, so also wives should submit to their husbands in everything.
Husbands, love your wives, just as Christ loved the church and gave Himself up for her to make her holy, cleansing her by the washing with water through the word, and to present her to Himself as a radiant church, without stain or wrinkle or any other blemish, but holy and blameless.
Boys are called to learn to lead, girls are called to learn to follow.
This is truth: the head of woman IS man in nature and order of how God created mankind, this is a differentiation of roles, leader and helper, not to put one down and the other up, but to free both men and women to operate in the freedom of their God-given roles. This is not a value statement. Our value to God and one another is identical. But God does Biblically call us to different responsibilities.
Colossians 3:20-21
Children, obey your parents in everything, for this pleases the Lord. Fathers, do not embitter your children, or they will become discouraged.