To CCNA: The Imperial City does have sewers but i guess they were made to keep rainwater at bay.
Clearly Skyrim is more realistic than you might think. The old nords didn't have bathrooms or toilets, not even outhouses. They used it the same way as the animals.
Rain sewers have been around for a long time. Romans even had them. It is just using them to carry away human waste was a new addition. I am sure the crews that carried away the night soil would use the existing sewers to dump stuff into. Pipes from a house into the sewer systems? That is a recent development.
The connection between human waste and disease was not made until the late 1800's with the beginnings of research into microbes. Up until then, people only knew it was a bad idea to poop upstream from where you drank, but not why it was a bad idea.