At it's heart, marriage is a passing of the authority over the woman from the father to the husband; usually for money. You see this authority of the father in 1 Cor 7 to deny marriage and in the various commands on marriage in Ex and Deut. You also see this in the adjectives used to describe forming marriage ('take', 'buy' etc). You also see in Deut 22:23-24 how a virgin women betrothed (i.e. he's paid the dowry for her but not yet taken possession) to a man is called his wife and sex with her by another carries the same penalty as with a wife. You see this in Deut 22:29 in that the man upon having sex with an un-betrothed virgin is required to pay the father dowry (if father approves) and 'she shall be his wife'.
Because money changes hands via dowry, it's not surprising a contract (or covenant) becomes involved. I'm not aware of any scriptural documentation of this occurring; but it was a cultural practice of the Semites going back to before Abraham. Ez 16:8 could buttress that but we're dealing with mixed metaphors there so it's not an open and shut case. Likely though, the presence/absence contract was the difference between marriage and concubinage. But in both cases they were possessed by the man; his women.
You also have Gen 2:24, 'leave and cleave'. Which isn't much different than the way we traditionally did marriage under English law before statutory law added government licenses. You were marriage if you lived together and held yourselves out as husband and wife. Additionally our traditional ceremony required the father to give the daughter to the man.
If you stop and think about this all, it's quite practical. It isn't necessary to over spiritualize this, create marriages where no one thinks they exist, nor heap burdensome requirements on top of something that is in every essence marriage.