It is biologically impossible for cousins to be identical.
Suppose that 2 male identical twins and 2 female identical twins marry. Then each set of parents will have identical genes. And the children of the 2 sets will be as genetically identical as siblings.
But siblings, except for rare cases of identical twins, have different genes from each other. it is impossible for 2 humans to have identical genes, unless one is cloned from the other with future science, or they are identical twins grown from a single fertilized egg which split into two very early in development.
Therefore anyone who cares enough about biological science will have to assume that The Patty Duke Show is fantasy or science fiction in which Patty and Cathy's mothers were both implanted with their identical offspring by magic or super advanced science.
Unless, of course, we assume that Patty and Cathy are not genetically IDENTICAL but are genetically SIMILAR ENOUGH in their genes determining appearance that they look almost identical. Just like Tsar Nicholas II and King/Emperor George V were not genetically identical but looked very similar and were hard to tell apart.
Thank you for one of the most intelligent post I have ever read!