It's all very well to declare that having never been in that position.
You asked me about my husband, not my entire relationship history. But why would anyone stay with anyone who wasn't committed to the same future plans, whatever that entailled (marriage, children, relocation, religion, house purchase...). It doesn't make sense. Why hang around being strung along? Just leave.
I don't see it as passivity. I don't think that having an expectation that a partner should show some commitment and character as being passive!
But it IS passive if it is always men who are expected to propose to women and women wait.
To me the lyrics of the song are primarily to do with signifying exclusivity and monogamy. Marriage is simply the way that most societies have established a settled, traditional representation of that mutual agreement between a couple.
But my point/OP isn't about the song. The phrase has developed a life independent of the song. In every example I've seen recently, the women have monogamy and cohabitation and sometimes a child or two. They are using the phrase about marriage specifically.
In a very old fashioned way, despite living very modern lives, but the language is demeaning in 21st century style. Quite the jumble.