This is absolutely a legitimate UX question! - Albeit more on the side of industrial and human factors design as it lacks a digital user interface (which is how many people define "user experience"). Nonetheless, it's always a great design exercise to study existing artifacts and how their affordance affects the way people will perceive or use them.
East Asian style tea cups don't need to have handles - if you take a closer look, the cups are generally made of thicker material (perhaps also makes etching/carving easier), especially the bottoms of the cups. This design may insulate the hot liquid better without use or need for the teacup's saucer plate. (If you look at the thinner material versions, I think you will often find that the top of the cup will have a "lip" that opens out slightly - probably also for making it easier to hold a hot cup.)
Culturally, I believe it's also considered more polite/traditional in Asia to serve/hold the cup with both hands. Right hand holds the top edge, left hand's fingers support the bottom.