There are actually some very real similarities. The ancient Romans were big on urban planning, to the extent that the word for those basic apartment buildings and the word for a standard city block were the same (insula, plural insulae). And the reason that the Romans built insulae in all new cities is because private housing was very expensive. This was in part due to the cost of materials (which means, of course, that public housing was typically made of cheaper materials). But if they wanted people to move into the new settlements, they had to make it relatively easy to do so.
They also had to make sure that this housing was in the center of the action since so much of ancient Roman life was conducted in public spaces. Wealthy people could afford to build their own houses outside of the public areas and travel into the main parts of the city. Some even had a separate house outside of the city walls. But poor and middle class folks needed to be more centrally located. In fact, the bottom floors of insulae were often reserved as commercial spaces for use by the people who lived above them.
There were also some important differences, of course. For one, a space in the insulae was free for early adopters as a way of encouraging people to settle there. For another, there was less segregation. Since both poor and middle class people lived in the insulae, there were fewer areas of concentrated poverty where only certain types of people lived. The wealthy could isolate themselves to a certain extent, but everyone else had to live together.
This was more true in theory than in practice, however. Over time, people would sort themselves into somewhat homogeneous clusters, just as we tend to do today. And unsurprisingly, the poorest were stuck living in the shoddiest buildings.
All of this is in contrast with individual housing in ancient Greece, which I now realize I forgot to mention. There wasn’t much in the way of public housing in ancient Greece. Instead, the ancient Greeks built their own homes (or personally arranged to have homes built for them). Housing was less expensive, however, due to the abundance of nearby materials. While marble was typically reserved for important public buildings, homes could still be made of wood and limestone.
In really ancient times, the houses were built mainly out of mud bricks with only their foundations being made of stone. But those would break down every few years, requiring people to constantly rebuild their houses. So as soon as it became feasible, people started building their entire homes out of stone.
Houses in ancient Greece were typically standalone buildings, though they were more likely to have conjoined walls in urban areas. And like the insulae, most of these homes would have a room set aside for whatever professional work its occupants did (e.g., a workshop). In urban areas, these workshops doubled as storefronts, and thus would be on the side of the house facing the street. Because just like ancient Roman life, ancient Greek life was mostly conducted in public spaces.
One big difference, however, is that ancient Greek homes typically only had one or two floors, and the whole building belonged to one family (though like I said above, two buildings might share a wall). The insulae could be up to nine stories high and housed many families (so again, just like modern apartment buildings). This made it much safer to live in an ancient Greek house than a Roman insulae. Since the living spaces in the insulae were all second floor and above, it wasn’t exactly easy to escape if the building started crumbling.
So yeah, the projects aren’t a bad comparison!