Designer Breeds

First the cockapoo. Then the labradoodle and the goldenpoo (or goldendoodle). And now there's the puggle, shipoo, pomchi and a whole mix of mashed breed names. All formerly known as mixed breeds or even , dare I say, mutts.
But that's not the point. What I'm curious about is why we have not adopted this euphemism into other realms of life. When a interracial couple have a child, could we not just make up a brand new 'race'/name? And it could extend to intercultural children, or children with a range of ethnic backgrounds. Let's embrace this idea and imagine how quaint the world would be! Designer children with designer ethnic hertiage!
But then the question remains, who gets to make up these fun and quirky names? And what then happens if two (separate) designer-bred children then produce their own offspring?