It's an adventitious root. I think that's a Hylocereus which is an epiphytic plant (grows on other plants). It's native to the jungles/rain forests of central america.
I recommend cutting both offsets off and planting them in some cactus soil. That's if you want three cactus instead of one. You can leave them, but the new shoots will take energy away from the graft (the red part).
You have a Gymnocalycium (scion) grafted onto the Hylocereus (stock). The Hylocereus provides photosynthesis services to the Gymnocalycium. The Gymno can't photosynthesize because it has been modified/bred to not have any chloroplasts. The red is what's left when you take the green away.