Like many religous conventions and customs marriage has, as its basis, a practical side. The idea that partners should "forsake all others" has a serious and practical effect. It means that the likliehood of incestuous relationships between, say, half-siblings, is considerably reduced. Such relationships have a greatly increased chance of producing deformed offspring and so in a monogamous society that risk is reduced.
With the popularity of marriage declining keeping track of the family lineage is considerably more tricky and the risk of incestous relationships occurring is increased. The answer for those wanting more than one partner is quite simple - just don't get married. So long as marriage is defined as a union between just two people (note the careful choice of words to avoid the same-sex marriage diversion) then those entering into it are entitled to legal protection should their partner illegally enter into a "marriage" with somebody else. Since almost all bigamists enter into their subsequent unions without the knowledge of their spouse such deception deserves to remain a criminal ofence.