Does this type of jewelry have a particular name? I am wondering if you would just call it "chain earrings". It seems too generic. Moreover, it's not just a earring, it's a nose ring too.
This is just called a nose chain. See its Wikipedia page:
Simply put, the nose chain is a link between a nose piercing and an ear piercing. Typically, these "chains" are just that: chain links, usually (though not always) made of some kind of metal. Yet, besides actual chains, the term "nose chain" can denote other types of connecting materials between nose and ear piercings, such as the common alternative of rosary beads. Other connectors can be used as well.