Why do people say this???

It came from the 18th century. Dogs were not used as pets back then but as helpers of the people in those times. Dogs are very hardworking and would do anything you tell them. If you want a better phrase, try this : You've been working harder that Donald Trump's hair spray! lol

As your astute observational skills have pointed out, to work like a dog is definitely inaccurate in most cases. While there is the unusual situation where dogs work hard, dogs who pull dog sleds for instance, most just eat, evacuate their bowels and bladder, sleep, and begin the cycle again! I feel that if one wanted to describe that another person was working very hard, they could say something like, "He was working harder than a one-legged man in an a&&-kicking contest." Now THAT would be someone working very diligently!