Engagement is the promise to marry another person. Marriage involves the legal commitment of a man and a woman to each other that is regulated by the government. Engagement is not marriage.
It is easy for people to conclude—wrongly—that engagement and marriage are the same when they look at the biblical account of Joseph and Mary (Matthew 1:18-25). At one point in time, “Mary was pledged to be married to Joseph” (Matthew 1:18). This is not speaking of the practice of engagement that we have. Joseph and Mary had given each other their word, by means of signed papers, that they would become husband and wife. That action legally established the marriage. That explains why Joseph had thoughts of “divorcing” (Matthew 1:19) Mary when he first learned of her pregnancy.
While Joseph and Mary were legally married by the signing of legal papers, they did not have the right to live together as husband and wife until the time of the public wedding celebration that took place days later.
The customs and legal nuances of marriage in first century Palestine were quite different from our practice of engagement.