I think grandparent was alluding to a separation of the two concepts of sexual relations and family structure: (Two or more) People can decide to bring up a child together without being in a romantic/sexual relationship. (Two or more) People can be in a romantic and/or sexual relationship without sharing responsibility of a child of one of the partners. Under this prerequisite, a family is any close social connection surrounding children (and the closeness and type of closeness being the linchpin of whether the term applies – not every flat share is a family for instance).

Regarding overpopulation: One motivation that I've heard quite often is that one of the most effective ways to reduce your carbon footprint is to have one child less. This is always presented with the asterisk that this is of course a very personal decision and there is the understanding that this is not a standard under which you will be measured. Still, if you're just looking at the numbers, that's what you get. Sharing parentage with more than two people is an alluring way to achieve the same result, especially as it means you will retain more of your personal time when child care is shouldered by three (or more) people.