Maybe to throw the scent off, manipulate a memory-woozy Woojin even more. If he was the beggar, then the childhood friend had a mole, so he's got to point the finger somewhere. I think those pills make Woojin more compliable either because they make him forget or because they make him more gullible so he believes whatever Hwon says. So, he may never find out whether his sister had that mole or not himself. Whoever he is though, Hwon is crazy obsessive....
what if he was lying about the sister having the mole and Woojin just added that to his memory after it was "suggested" to him that it was real
The author threw that last bit in with “i was so happy when you said it didn’t matter who I was” making people think ohhh could be the beggar but how would the beggar have know his sister had a mole under her eyebrow? I think the reasoning for the author saying that is to further show the obsession. In his mind he’s thinking “oh he loves me no matter who i am” my theory