In "The Patty Duke Show" the premise is that there are two identical cousins; not siblings: cousins.

I scratch my head in befuddlement wondering about the biology which led to this result.

It's a stretch, but the closest I can come to a quasi-plausible explanation for identical cousins is that the same guy impregnated identical twin sisters, or in the alternative the same woman was impregnated by identical twin brothers.