I agree with Stefan. 100% sure it is an Echinocereus. This one is going to be hard to repot without incurring some damage to both the plant and most likely you.
The main issue is that the plant has become so sprawled that getting soil passed it to fill up the new pot will be hard as you will not want to give it too big of a new pot - in a way the small pot has probably helped you avoid giving this plant rot - most Echinocereus are very sensitive to being wet too long. In this case a small pot probably made it dry out pretty fast. So while increasing the pot size at this point seems prudent, be careful to not go too large, or if you do be very careful with watering and check thoroughly how fast the new pot dries out.
I use salad tongs (repurposed as cactus grabbers - if they have any kind of softer pads on them there is not going back to using them as salad tongs after using them as cactus grabbers) to grab onto plants like these to lift them etc. I suspect that you will probably knock some pieces off regardless of your method, that may actually help you get soil around it.
While I agree with Stefan, that they are hard to reroot, it is not impossible. These sections look pretty vigorous so if you do end up with some pieces that came of the plant just still them in a pot with soil and essentially do not water them till you see new growth. Some will shrivel up and die, but almost certainly you will get some too root and survive - if you are interested in getting more than one of these that is.