The simple answer is Spectacular Engineering. The complex answer is . . . more complex. Though it seems that both figures are supported by a boom under Belle's dress - a boom which is able to track forward and backward on a sled, and that whole sled is on a platform that rotates, and the "floor" they're "on" rotates with that, but is painted so it's not noticeable that the floor is rotating. The Prince's legs are then basically free to be manipulated at the thigh, since he is not actually supported by them but instead secretly cantilevered by Belle. In essence, where typically Animatronics have their power run though their feet to power their movements, The Prince's power is supplied somewhere around his hips and kept aloft.
Mechanically speaking, the figures are so foundationally integrated with eachother that it is likely the device that manipulates them would technically be considered one Animatronic that is simply designed to look like two, you know?