The allusion is to a butcher's dog, which would be expected to be very well fed from scraps. Why that is considered to epitomize fitness isn't clear, as it might be thought more likely that the dog would be overweight than fit. John Camden Hotten, in A Dictionary of Modern Slang, Cant and Vulgar Words, 1859, defined 'butcher's dog' this way:
"To be like a butcher's dog, i.e. lie by the beef without touching it; a simile often applicable to married men."
That's clearly a different meaning, i.e. butcher's dog was then a metaphor for 'something we are close to but cannot have'. That meaning has gone out of use.