woah that story sounds crazy, so like it could give you memories and feelings that aren't yours but the previous owner?
There have been instances of that, yes. Obviously it doesn't happen all the time or more people would be aware of that possibility. Most people tend to brush it off as fiction or "psychological" responses, but there was no denying what that little girl saw in her nightmares and their correlation with her "new" heart's former owner.
Oh yes I remember reading this story of a girl who got a heart transplant and after the surgery, she begin running and picking up hobbies she never did before. Turns out her former heart owner was a runner too. So I agree, the dream of his dad isn’t entirely impossible
Although it can't be proven [as of yet], that doesn't mean it isn't the truth. To brush it off as it simply being "psychological" feels really dismissive and uncaring towards the MC's worries/feelings:
http://www.mangago.zone/read-manga/a_loving_existence/uu/d_chapter-5/pg-4/
I remember a true story I heard many years ago about a young child who had a heart transplant. The little girl started to get violent nightmares of being stabbed by an unknown man every night. It got so bad that the girl's mother started to research whose heart that her little girl had received. Of course, with the patient privacy on the line, it was very difficult for the mother to get answers. But, eventually, I think it was one of the deceased's family members that decided to come forward. The heart the little girl received was a murder victim that had been stabbed to death by an unknown [male] assailant. I don't remember if the stabber had been arrested or not.
But I have heard and read of many other similar situations. Our bodies store deeply emotional and or severely traumatic memories, even after death (for a period of time). It's not impossible for the heart's former owner to have visited the MC in the dream if that was his desire—one that maybe he wished for with all his heart—to convey that his son needed love and support. But these are just my thoughts on the matter.
On the other hand, I absolutely adore this concept and I look forward to reading until the end; whenever that may be.