Manhattan's population was 2.3mm in 1910 but is only 1.6mm now. Where did everyone used to live? Surely there is more housing stock/volume now. Did people simply average much more cramped quarters a century ago?
I can't get you all the way to 1910, but I can give you some insight back to 1940.
The census bureau has kept track of overcrowding since then.
They define overcrowded as more than 1.01 persons per room (If you have an apartment with one bedroom, a kitchen and a living room, more than three people live there) and severe overcrowding as more than 1.51 persons per room.
In 1940 20.2% of all housing units in the US had more than 1.01 persons per room and 9% were severely overcrowded.
A report that covers 1940-2000 can be found here.
The current (2016) ACS estimates put those numbers at 3.4% and 1.1%, respectively.
If you wanted to drill the data down to just Manhattan you can dig around at https://usa.ipums.org/usa/ and https://factfinder.census.gov/faces/nav/jsf/pages/index.xhtml but I think your guess is correct.
We were content to be a lot more cramped, back then.