Can any species be bred selectively/engineered to become as diverse looking as dogs?
Dogs have a genomic structure that allows breeding with high variation in size, shape, coat quality, color and other qualities particular to each breed as well.  

Other domesticated animals can be bred for as many qualities, but dogs in particular show a wide level of morphological traits - varying in size from just over a pound to the size of a wolf, from which dogs are derived and genetically are still compatible.  But more interesting than just size or coat color/texture and even their intelligence and personalities, the proportions of their bodies, of their skull length and breadth, are remarkable.

There are over 160 registered breeds of dogs, but this is only a measure of how much time people have put into them.  I think its possible to get nearly anything you want with animals, if you are patient enough - its not clear what is and is not possible with enough genetic manipulating. For instance, horses can be bred over nearly as great a size range for instance (the miniature horse the size of a large dog, the Shire is 3,300 pounds), but it would not be as easy to get both the size and muscularity and shape of a bulldog in a horse.  Breeding a mouse of various colors can be done, and so can interesting behaviors, but body shape seems to be harder: a Weimaraner mouse could take a tremendous amount of time and animals.