I'm going to go way out on a limb It may be a broken dowel extractor. The square cut-out in the handle was used to drive the tapered piece into the broken off dowel (like the end of a chair cross-brace pegged into the leg) and then the handle screwed down to pull the dowel piece out.
Have not seen one like that but I have seen such tools that were similar. Lots of furniture used to be dowel/tenoned together