Question

What was the name given to low-cost, multifamily dwellings in the cities in the late 1800s?

Answer 1

I think the name you are looking for is "tenement". 

The name comes from tenancy: the flats in those dwellings were typically rented, not owned. 

Tenements bring with them the association of being overcrowded and having sanitary problems. 

Tenements were an important aspect of life in New York in the 19th century.