I am posting my reply as an "answer" because of its length and detail, but credit goes to Nate Eldredge and Andy Gillett (see below).
I agree with Nate Eldredge's comment (May 5 at 18:38):
I always assumed that APA saw the need for a standard style and
created one, and then other fields simply adopted APA's style because
it was good enough and less trouble than creating their own.
An English professor and professional writer (Andy Gillett) published a superb blog post in 2014 that provides the most thorough history of "APA Style" I've seen. The blog post is titled, "'The Harvard System' of referencing" (September 28, 2014).
Although the post's primary subject is the so-called "Harvard System", Gillett discusses APA Style at some length. For example:
The earliest version of the Publication Manual of
the American Psychological Association was a short, seven page guide
published in the Psychological Bulletin in 1929. This was followed by
a 32 page version by John E. Anderson and Willard L. Valentine in a
1944 issue of that journal. The first edition of the Publication
Manual appeared as a supplement to the Bulletin in 1952 and was 61
pages in length. It was revised twice prior to the publication of the
second edition (1974, with many reprintings).
Gillett also offers a nicely detailed list of the various iterations of APA Style. Here is the beginning of that list:
History of the APA Publication Manual
Instructions in regard to preparation of manuscript. (1929). Psychological Bulletin, 26, 57-63.
Anderson, J. E., & Valentine, W. L. (1944).The preparation of articles for publication in the journals of the American Psychological
Association. Psychological Bulletin, 41, 345-376.
American Psychological Association, Council of Editors. (1952). Publication manual of the American Psychological Association. > Psychological Bulletin, 49(Suppl., Pt. 2), 389-449.
Although Gillett's blog post doesn't necessarily provide a complete answer to question posed by TrivialCase, it provides more history, detailed information, and references than anything else I've been able to find.