If you're a member, you belong to a group or club. The late comedian Groucho Marx famously said, "I refuse to join any club that would have me as a member." Go figure.
When an object, like a plant or an element, is a member, it's also part of a group or class. A tiger, for example, is a member of the genus called Panthera, and it's also a member of the cat family. You're a member of the human species, and if you swim competitively, you're also a member of a swim team. The United States is a member, along with 192 other countries, of the United Nations. The Latin root of member is membrum, "part," and also "limb or part of the body."