Technically they are both types of rayon manufactured from bamboo and birch trees, respectively. They are both synthetic - because they’re tree-derived but processed into fibers - but if you search for rayon you’re going to get a lot more than those two types of fabrics.

You could also look up TENCEL, the name of a trademarked fiber material used in modal fabrics. It’s present in a lot of different types of textiles beyond clothes, but they are great for underwear. I like modal a little better than bamboo, but it’s good to try different materials to see what’s right for your body.