Effectively, for all practical purposes - yes.
Let's say you inherited an axe. It came from your father, who got it from his father, who got it from his father, and so forth. It's always been considered a family heirloom. However, you find out one day in intensive research of old documents, that your great-great-great-great-grandfather actually stole it - let's say he bought it from the local store on credit but never paid. It would be completely impossible to track down the descendants of the store owner even if there are any - and if there are, there are probably thousands of them by now, so if you were to fairly give it back you'd have to cut it into tiny pieces.
Is it your axe?
My answer would be simply "yes". It is your axe, now. It shouldn't have become your axe. But it did, and is now your axe. Your axe just happens to have a fascinating history.
So we'll disagree about this, because I would say that the axe continues to belong to the person from whom it was stolen. That is would be difficult to trace records to discover the actual owner would not change its ownership, or are you asserting that it started belonging to a descendant of the thief the moment it became difficult to track down the descendants of the victim? It's not just a fascinating history; it has a grifter history.
I know this makes me a stickler of sorts, but I would assert that the horse the Frenchman ate (horse is mighty tasty, by the way) continued to belong to the person from whom it was stolen even after it was consumed. Possession and ownership are simply two different concepts.
And here is one of my intellectual justifications for maintaining this position: if one ever knows what to do to successfully raise a young man to become a life-long adult thief, follow this prescription: starting as a little boy, when he steals something, only require him to pay for a replacement. The lesson that is absorbed is that the worst that one has to do is pay for something one fully utilized, but every time one isn't caught one got something for free. The Grifter's Creed. The only way to break a stealing habit is to create multi-layered multiple punishments. Look at what YHWH asserts in Exodus 22:16-17: not only does the daughter-thief have to pay the bride price, he has to throw himself at the mercy of the father if he wants to continue the relationship in legitimacy, and if he is refused he goes away empty-handed unless he conspires with the daughter to run away with him. In essence, though, in most cases of refusal, the thief stole a man's daughter's virginity, which can't be replaced, but he had to pay the price that would have been paid not only for her virginity but for a lifetime of helpmeeting from her.
In our American justice system, in almost every case in which all that is required of a thief is to pay just compensation, the system is terribly ineffective at preventing future theft by the thief or discouraging others to become thieves.
In the example I gave, does the woman in question belong to the man in question?
Would she say of him "this is my man"?
Would he say of her "this is my woman"?
If you're talking about the example of the man and woman who have completed a contract but haven't yet (or perhaps ever) consummated their relationship, then, yes, the woman belongs (is possessed by) the man in question, and they may call each other their man and woman, but that's just vocal gas. They can call themselves King and Queen or Frog and Toad for all I care. The average person would at the very least say, that's not a real marriage, if they perpetuated this framework for years and years.
Let's say they are together for 10 years and have half a dozen children. Then look at the scripture and decide they were wrong to elope without her father's blessing. And, for arguments sake, let's say he still disapproves and has not changed his mind.
What would you have them do now?
I would have them stay together. As I'm indicating by my own intentions, I would consider the optimum thing to do to be belatedly asking for the father's blessing if he's still in the picture. If the father refused to do so, though, what I would have them do then is continue to be married, taking care of all aspects of family business to the best of their ability and in as close to conformance to Scripture as possible.
I would no more have them split up than I would do what missionaries do with polygamists in Africa.