The saying "man and wife" is related to the fact that in previous generations, women were not viewed as fully autonomous individuals, but rather, were defined by the relationships with men in their lives. The use of "man and wife" within a wedding ceremony allowed a man to retain his own personhood, but transformed a woman into a "wife", which implies that her role as a man's spouse supersedes her own personhood.
That being said, many modern wedding ceremonies do actually use the phrasing "husband and wife" instead, as overt sexism has diminished and couples prefer to demonstrate equality over using the traditional phrase.