Comment by Yasi Kuro on At the End of the Road - Chapter 45

I was also confused the first time I read this, lmao. That's why Taemin (MC who's now in the body if Han Siwon) calls Siwoon (ML) Min Woojin, coz that's his real name. 

Min Woojin = Siwoon (ML) pretending to be the legitimate son
Taemin = Siwon (MC) woke up in a different body

So basically, they live using different names now, but totally in a different situation.