Was there a person like a mortician whose task was to dispose of dead people who would not have funerals? How did they dispose of the bodies?
There's a famous unidentified body from 19th century Paris--a beautiful young woman known only as "L'inconnue de la Seine" (the unknown of the Seine River). Her body was recovered and the mortician made a face mask of her so that she could be identified more easily--looking at a mask is a lot more appealing than looking at a rotting dead body. Because of her beauty, she drew a lot of attention, but in some other cases, they also did molds of the face. Once photography was available, photos might be taken of the unidentified after they were buried. In this case, though, she was never identified, but her face became a popular work of art, and the Resuscitation doll used in Red Cross classes on CPR and mouth-to-mouth resuscitation, now has her face.
In Paris they were displayed in a sort of underground morgue for relatives to find and identify, if any. If not claimed they were buried in a mass grave/burial pit that was kept open until full.