Typical 3/8" male drive with spring loaded ball for retaining a socket.....BUT....has a male 5/16" square drive (no retention device).

No markings, seems cheaply machined, with coarse machine marks and thinly chrome plated.

It came in a Kraeuter metal cased 3/8" metric socket set, but clearly does not match those pieces.

Huh?

Something like a pipe plug adaptor perhaps, male drive would be a bit odd tho...

Maybe early drain plug adapter, say 1910's to 1920's.

Many old cars had a rear axle differential drain plug that you used a male plug wrench to take it out with.
No external square for a crescent wrench.
When I worked in service stations during the 1950's, this was what I used.