Duplexes in the US can be split either horizontally or vertically. I live in a duplex--I have the entire second floor, plus a laundry room and storage area in the basement.
The house next door to mine is also a duplex, but split vertically.
In and around Boston, there are tri-plexes. I have no idea what the correct name for them is, as everyone just calls them "triple deckers." They are a three-family home with each "house" being one complete floor of the building.
How do they access their house? It looks a bit like the large terraces here which are split into flats or purpose built flats which often have communal hallways.
If you look carefully at the front of the house under the porch, on the ground floor, you will see two front doors. One door will lead directly into the ground floor house, and the other will lead to a stairway that will allow the residents of the other floors entry.
I think the reason these weren't called apartments back in the day was that a good many of these buildings were owned by one of the residents. They'd either house family members on the other floors, or rent the other floors out. But they didn't have the feeling of an apartment, because of the separate entrances and they had windows on all four sides and because they were on their own tiny plot of land.
Most of these houses would have porches on the back of the house, with another stairway, either inside or outside. The porch would be used for hanging out laundry, and the back stairs would give the ice man a way to get the blocks of ice into the kitchens without it dripping all over the place.
Another variation on these was a two family house, with the ground floor being one "house" and the top two floors being the second "house." I lived in one of these for a while, but the top, or attic, floor had been made into its own space, being rented out as an apartment.