The elliptical FRP–concrete–steel double-skin tubular column (DSTC) is a promising compression member due to its favorable mechanical performance and constructability. To provide insights into the interaction mechanism of the individual parts (i.e., elliptical FRP tube, elliptical steel tube, and sandwiched concrete), the compressive behavior of elliptical DSTC stub columns was systematically studied and compared with two types of reference specimens, including FRP-confined solid concrete (FCSC) and FRP-confined hollow concrete (FCHC) specimens. Based on a comparative investigation on the compressive performance and dilation properties of concrete cores in different column forms, a three-stage interaction mechanism in elliptical DSTC was clarified: unconfined stage, transition stage, and hardening stage. The confined concrete strength in elliptical DSTC increased by 20%–55% compared with the corresponding elliptical FCSC with the same column dimension and FRP confinement level. The possible reason is that the less-effective confinement zone in elliptical FCSC was replaced by a hollow steel tube. The existing model could not be directly adopted to predict the behavior of elliptical DSTC using high-strength concrete.