The lyrics of the song describe a bizzare, but plausible scenario, in which a man finds himself becoming his own grandfather through an unfortunate combination of marriages. (The narrator marries a widow, whose adult daughter then ends up marrying his father.)
Thinking back on the times that I've heard one person mock another by suggesting that their lineage contains such an anomaly, this was always done by stating the specific relationships. For instance, the person would say something along the lines of "I bet [this person]'s brother is his uncle".
That made me wonder... Has there ever been an actual word to describe the state of being related to yourself as a result of a marriage?