Part of the problem is that we've invented this term marriage that doesn't exist in the Bible. Neither do the terms husband or wife. The Bible only refers to men and the women they've mastered. Yes, you can make a covenant with a woman you've mastered if you like. It's not necessary though.
I’m not sure the covenant has to be “with” her. God makes covenants throughout scripture based on his own will and in many instances (possibly all) the other party is simply informed of the covenant.
I am actually starting to lean more towards the “both” position and the covenant doesn’t have to be formal or even spoken. It can simply be a mindset that the man has when he takes or purchases his woman.