The induction coil in your box looks, if not identical, at least very similar to the induction coil in an Automatic Electric RR subset that I have, which was used to connect a scissor-type transmitter arm, or a desk stand, to RR lines.
Here is a view of the induction coil, seen just behind the condenser in the front. It is mounted in one of those small Type 32 Bakelite extension boxes and has a push-to-talk button. The boxes already showed up the AE catalogs in the 1930s, for various applications, including as relay boxes, small ringers, and local extensions.
I have not found much RR equipment documentation for AE stuff either, and I am always looking. Railroad telephone technology changed very slowly, and what would be considered archaic systems still existed in abundance in many places. A friend told me that in some old subway station in NYC scissor-type transmitter arms could be observed in some old control booths even in the 1990s, albeit probably unused by then. I have a WECo 1348D RR transmitter arm for local battery service that was still refurbished with new parts (bull-dog transmitter) in 1955. Same situation in Europe. I have a 1962 Yugoslavian (! Tito era) magneto phone in mint condition, never used apparently, which was a copy of a 1933 German local battery telephone, the OB33. Even in Germany, those were still to be observed in the field until the 1990s, I was told or read somewhere.
There is some good documentation, in the TCI library and elsewhere, about the Western Electric RR systems. The technology didn't change much over the years. A 501A subset pretty much was the same in 1949 as it was in 1924, albeit having been renamed as 501E, IIRC. I think they changed the paint. :-)