Tree branch muddler
This object sparked some interest in the community because Tiffany & Co. is mainly known for making jewelry. It’s pretty clear from its appearance and the OP’s description that this was not exactly a piece you could wear around your neck. However, the answer to the question was answered within the first couple of posts.
This thing with a rounded end is called a tree branch muddler, which is mainly known as a muddler nowadays. You use the rounded end to smash fruits and other things at the bottom of jugs to release their flavor. Bartenders often use muddlers to make exotic cocktails.