I also think that it is his mother singing (A lullaby perhaps?).
What makes me think this is that during the flash back when he was at his home, he thought about a time where his sister told him that once he is older she would go with him to find their mother. If the topic was brought up it is definitely something that Hyun Su wants or craves (motherly love).
Then later when the shopkeeper thought about the past, he saw Hyun Su listening to the tape, he was smiling peacefully to himself. We know that the ML presumably has Antisocial Personality Disorder, so isn't this a bit strange? People with ASPD usually struggle to keep meaningful relationships but the person in the recording seems to be someone he cares deeply for. Enough to lash out and become aggressive once an object reminding him of person is out of his possession.
But I would like to talk about something separate also. We are told that he would NEVER separate from the cassette because he is so possessive over it. But didn't he leave it behind? He left it behind after he found the FL, Ji Won...this has lead me to believe that the FL is much more important to him than he thinks. He has said that the FL keeps his father away (But of course this is only in his head). Due to him leaving the cassette when he met the FL, it has made me think that while is father was alive the cassette of the lullaby may have kept him at peace and gave him a 'safe place' in a way. Maybe the FL is now that for him and he no longer needs the cassette for that reason. I just think that due to his inability to process emotion/feelings, he is struggling figure the importance of his wife to him (I personally do think that he holds feelings for her).