How does it work with parents having a child? In a sense that child is married to two adults, both of whom are responsible.The problem is that part of the modern definition of marriage is that it marks the person who is most important and closest to you. It gives them primacy. How does this work with two people?
Really, what you're driving at is nothing more than minor contractual issues. Partnerships are formed every single day. Businesses seem to function just fine and dandy with all sorts of partnership arrangements.
Working out the logistics would not be difficult. Hell, you could leave the statutory rights out of it and simply require the filing of partnership agreements just like we already do for, well, partnerships.