That may sound strange at first, the names of caves are used more than once? A name like Sand Cave, Bat Cave or Herrmanshöhle exists several times? But it’s not that surprising, we're used to cities being given additional names because otherwise the names are not unique, like San José, California and San José, Costa Rica. Even "Old" York and New York are not unique, there is a York in Alabama, Arizona, Illinois, Maine, Nebraska, North Dakota, New York, Pennsylvania, South Carolina, and four(!) in Wisconsin. In the case of caves, there is another point: some caves have characteristics that have led to their name, which are rather common in caves. The prime example is probably the bat cave. There are bats in many caves, the number of "bat caves" is legion, often you only realize this at second glance because they have different names in the respective national language. And then there are caves named after the same Saint, like dozens of Saint Michael Caves. The Chinese have many famous fairy tales and legends, and everything is named after them, so there are plenty of Yellow Dragons and White Clouds.
So why is this an issue now? Well, when you arrange to meet up with friends at the Bat Cave and one of them is in Kentucky, the other in Ontario, and the third on the film set of the Batman film in Hollywood, it gets a bit annoying. The usual strategy to solve the problem is, of course, the same as with place names: You append something else. Usually the name of the city, the region or some local proper name is added. In China, where there are many names that are used several times, adding the province has become common practice.