I ran across this at a garage sale today. The wife and I must have stopped at over 20 sales and had found nothing, then we hit this sale. While not great,it did yield some jaws for an Eifel plier-wrench in the old canvas pouch along with this strange vise, and another tool that I will post later.
The piece is marked VISE JOCKEY, reg. trade mark, U.S. Patent No. 2,595,699. The patent was issued May 6, 1952 to a Carl E. Petersen of Brooklyn, NY. It is an extremely well made piece, but I had never seen one and had no idea how it was used. I was really lucky that there was a patent no. on it. While not in DATAMP it came up quickly in Google Patents.
It is a precision vise that was designed to be held in the jaws of a regular bench vise. I tried it in a little bench vise to see how it worked. I was not impressed. It looks like a solution in search of a problem to me. Maybe that explains why they don't appear to be real common.