The number has been around for a long time.
For example, there is a book published in 1949:
Practical Poultry Management, 5th Edition
by James E. Rice and Harold E Botsford
I found a .pdf of it at the internet archive:
https://ia801806.us.archive.org/17/items/in.ernet.dli.2015.233717/2015.233717.Practical-Poultry.pdf
(Hopefully that link works. Otherwise, try googling for it by title and you might be able to find it.)
On page 89:
"The smaller the flock, the more floor space is required per hen. Ten hens might need 6 to 8 square feet per bird. Leghorn hens may be kept profitably in flocks of 100 or more with an allowance of 3 square feet per hen. For heavier varieties, 4 square feet per bird is the usual figure. It is always well not to overcrowd."
I have seen the same numbers in other books published at various times over the last century or so. I sometimes see bantams mentioned as needing only 2 square feet each.
I don't know if there have been any formal studies done or not, but I would expect them to have happened quite a few years ago if they were. Studies that old can be difficult to find online (I've seen some on other aspects of poultry raising, but haven't specifically searched for ones on square footage.)
Like any other rule of thumb, it will work well enough in many cases but have problems sometimes.