Due to the fact that it can alter time and history and The Red Guy had been playing around with it, the Primordial Soup from the episode, Time Machine, is the reason why there are supposedly multiple Red Guys.
Cow's birth cow was a surrogate for Mom and Dad. This is one of those universes where humans and talking animals coexist without question (a la Get Fuzzy); Mom and Dad are animal lovers (as are their family), so they decided to have animal children instead of human children. Chicken, they hatched from an egg, but Cow was arranged through a surrogate cow - this explains Red Guy's flashback of him delivering, "Mom's baby," even though she was out in the waiting with Dad the whole time.
Red threatened to wash Chicken's mouth with soup. Now soap may sound more cruel, but what if the soup used had chicken stock (or chicken broth) in it?
This is another case of Furry Confusion: in a bumper for Cartoon Network, Chicken confesses to eating beef when looped into Wonder Woman's lasso of truth, not to mention Cow has been seen eating poultry before. On top of that, both Cow and Chicken frequently eat pork products, whether in the form of weenies, sausages, or ham... and they have a cousin who's a sow.