Here are pictures of the trap and the hole.
"I am not sure how they made that connection in such a short space, but if there is a trap below it, then that may be some handyman's idea of a way to connect a sink drain to the sewer system. "
There is no trap below. This is the trap and it isn't a handyman special. It is a legitamate plumbing "fixture" from 40+ years ago. It works fine, except that it tends to collect gunk in the bottom that eventually fills up the trap to a level above the outlet. Then you have a problem, since there is no way to get a snake through it to clear it. Thus the invention of the P trap, I would guess. Another one of these I saw had a cleanout at the bottom. Good for a crawl space, bad for under slab.