I can't help you, I have not encountered a tub or shower drain that looks quite like that.
That said, I did just change the drain on a sink and the piece you screwed out looks an awful lot like the chunk of metal that sits in the bottom of the sink. If I had to guess, and this is strictly a guess, the piece you took out is analogous to the one in the sink, and the part of the drain underneath is something like an elbow with matching threads that screws onto that piece. It looks like there is a rubber seal on the bottom of the tub, above the elbow (or whatever shape it is), which seals the whole thing, when the two pieces are screwed together.
Try looking carefully all over the piece you screwed out for some sort of identifying markings. If that doesn't work, take it to a plumbing supply store that has been around a long time and see if they can identify it.
If you can obtain a replacement piece you might be able to just screw it in.
But...
I have doubts that the existing rubber gasket will seal again. It has been compressed for who knows how many years, and is now released. It may be too rigid and hard to seal properly again. To replace it you probably have to get under the tub.