After doing a lot of (possibly not quite legal) digging, I think I've found the rough birth of the problem.
The problem seems to have evolved in around 1940, from the M.I.T campus during World War II.
The exact author appears to be unknown.
An early use of the problem was in 1981 in Rick Billstein's 'A problem solving approach to mathematics for elementary school teachers', in which Rick Billstein uses the census-taker problem when in Chapter 4, discussing 'Integers and Number Theory'
On page 216, the solution process to the census-taker problem is broken down step by step, but there is no indication of where the puzzle was from, whether original or sourced.
(Page 215 contains a small paragraph at the bottom that gives the problem, but also does not mention a source).
Looking around, I found this from 2010 that revisits the Census-Taker problem, and within it, on page 2 it says:
, the census-taker problem originated during World War II. In
literature, it has appeared in different stories in a puzzle-ese style. As a result,
many number-theoretic problems with seemingly insufficient information have been
proposed in some journals.'
In the references,
 is 'M.H. Greenbelt, Mathematical Entertainments, Crowell, 1965.' Doing a bit more digging, it appears this is a book from 1965:
Sadly from here, I have been unable to find anywhere which fully disclosed the contents. However, what I did manage to find is this google books page which shows small snippets of the paragraph containing a key word. Trying several key words eventually led me to this snippet from Page 1:
'One of few amusing things to come out of World War II was a new type of brain twister - the "census-taker" problem. (The time and place of a problem are difficult to specify. To the best of the author's knowledge, this problem was born on the M.I.T Campus in one of the war...'
I have been unable to find a snippet of the second half of the paragraph but I will keep trying.
Finally some google ngrams, showing the use of 'census taker' over time in books:
You can see there is a large spike in both in 1940, which seems to be the birth of the problem, and also in about 1970 which seems to be when it started to be published a bit more.
Overall it appears that the original problem came from the Campus of the Massachusetts Institure of Technology during the early 1940s.
However, the problem itself became very popular in the 1980s, appearing in many mathematical writings, and again in the 2000s, being asked about on many forums and still is to this day. Notably, the earliest forum post involving this problem that I could find was here in 1995, where it was answered a year later.
For similar puzzles, you can look for 'Number theory riddles' online, and the majority of the ones you find are likely to be classic problems as well. The census-taker problem seems to be the original, which gave birth to many other ideas for other problems using this idea.
Hope this helps! It was fun to research.