How are families defined in The Giver? Are families the foundations of a society, or are they continually open for new definitions?

Families in The Giver are indeed the foundations of society, but these families are wholly artificial constructs, designed by the community's Elders to satisfy specific social needs. In that sense, the family unit exists purely and solely to serve the interests of the state. And what the state demands most of all is stability.
In constructing the building-blocks of society, then, nothing can be left to chance. This means that every minute aspect of family life must be controlled and monitored by the state. People do not just spontaneously fall in love and start families as they do in normal societies. In the community of The Giver, it is the government that decides who gets married to whom. It is also the government that entrusts the raising of children to specially selected Nurturers, such as Jonas' father, who take care of newborn babies once they've been taken from their Birthmothers.
Under this rigid system, there is no room whatsoever for any difference in family structure. The government will only permit the kind of family unit that we've already been discussing. Any other kind might undermine the government's control over the foundations of society, and that's a risk that the community Elders are simply not willing to take.