Here's an unusual one

Usually duplicate sets are not hermaphroditic. This is most often a fused twin, called a Chimera after the mythological beast sharing the same name. It's like an extreme form of Siamese twins. There are many human cases, one of the first noticed being a woman who had children that would've belonged to her sister except that her sister was not born but rather incorporated into her own body. The "sister" contributed the ovaries. Genetic screening was done, her children almost removed from her house because it appeared as though she had kidnapped them from a relative. Eventually of course it was discovered that she was a Chimera and now this discreet form of "twin" is well recognized. Often it only is manifested in a patch of discolored skin, like a large birth mark, but sometimes there are dramatic examples such as your doubly endowed dog.

Not to send this straight into the gutter, but along the lines of fascinating genital conditions- anyone know of the Guevedoces? This is a real hermaphroditic (or psuedo-) condition where children are born phenotypically female right down the genitalia until puberty at which point they begin to produce enough androgyns to "grow" male sexual organs- in reality they are descending and expanding as they normally would have in utero of course. It's most common in certain Carribean populations, hence the Spanish Guevedoce meaning "penis at twelve." After puberty they are generally normal, heterosexual males except they are sterile as a result of having the testicles too long inside the body exposed to higher temperatures than are tolerated.

Sounds very... complicated.

Very rare globally, just like the dog. Really interesting, thanks for sharing.