I've actually never used this saying because most dogs I've ever known were well taken care and either never got sick or was immediately taken to the vet, so I decided to search for the origin of the saying "sick as a dog."
And found nothing. The only things I could find pointed to the saying originating in the 1700s and may come from the fact that since dogs eat everything they find, they tend to get very sick, or that back then dogs weren't well taken care of and thus when they got sick, they got really sick, or that back then dogs were given food that was not fit for human consumption thus ending with them getting sick. Either way, Kewpie is very insulted by the saying which takes dog progress back 300 years.