There are cases where a person can develop a split personality disorder or bipolar disorder due to severe trauma ( like in the case of Dr. Jekyll and Mr. Hyde), which is most likely the case for Lee Muhan. The three Mugos are so different in personalities and their behaviour, but they're also similar in some aspects.
Now, since Lee Muhan was killed to make a "god" , he's basically dead, but his soul probably couldn't rest, so the three personalities of Mugo were formed. They also mentioned that he was cursed, which is also a probability that we can make as to why his soul hasn't rested.
This is just my opinion tho( ̄∇ ̄")

Im lacking Brain cells i messed up the story so who is he if he isn't lord mugos and who are lord mugos i got lore questions but they not coming rn it's 3am in the morning