Wurm Online has animal breeding right now. It is a bit silly but you lead one animal to another and while leading click on the other then Breed. "They get frisky" is the only wording you see. If you succeed the female is pregnant, keep her health and fed and after X period of time a baby appears. Simple and traits carry from previous generations. So you can play breed the best dog or fastest horse or whatever.

I do not know of any other game that does this. At least not any other MMO. So it is possible. One man can code it like ol Rolf.