You may want to go back and look for the stopper. It would be glass and likely with C&CO. or the monogram, I'm not sure which.They are cute little things.
Something like that but funny thing is, I don't think that goes to your bottle. The neck on the bottle is ground and should take a ground stopper to match. That one looks threaded.PS, the bottom isn't broken off. Well it is but it's made that way so not in a damaged way.
It's a general type used and Worcestershire and other sauce are most common but whiskey and other bottles too. Sometimes they're embossed with the name. They would have had cork wrapped around the stems.