What is this four-outlet ungrounded receptacle called?
Ed didn't really seem to answer your question, other than the 'why' so I did some digging and found this a page that references the polarized version of the socket (see #5).

According to the page it is called: 


  4-plug socket


And 


  dates back to mid 1950s


Because the version in your picture probably when out of production just before the one linked (polarized one) came into production we can assume this is the end date of your particular yoke.

However, as Ed said, the last of the "4-plug socket" style yokes likely went out of production when the 3 prong grounding requirements came in.

As a side note: I've had experience trying to plug things into the Monolite Quintet socket, and found that even though it was designed for polarized plugs, one generally cannot plug two polarized plugs in next together at such a close proximity due to the fairly common skirt on the polarized plug.  The "4-plug socket" seems to have similar spacing.