I think Inho keeping Gihun’s bloody tracksuit jacket unwashed for six months and then giving it to Gayeong is perfectly in character for him.
His obsession with Gihun was always unhealthy and filled with such intensity, Gihun became the center of everything that he did, all of his choices, thoughts and feelings revolved around Gihun for this time span, their ideological battle but also Gihun as a person consuming him whole. (Assuming that it started even before Gihun rejoined the games, because he already showed curiosity about him when he saw him winning, so after their phone call in the end of s1 when he saw that Gihun wouldn’t let go so easily, he most likely tracked every of his steps until they met again.)
When he lost Gihun all of this was suddenly gone in a matter of no time, leaving him with an even bigger emptiness than he ever felt before.
So him clinging to that jacket and trying to mantain as much of what’s left of Gihun’s presence makes perfect sense, it was his way of grieving and showed how attached he really felt towards him. Underneath his deep layered attachment there was also a tenderness, something that expressed his devotion to Gihun and his admiration for everything that he did.
Inho thought giving this piece of clothing to Gayeong was the right to do and that showed how much he’s still disconnected from reality. What would seem cruel and sick to others, would feel like a good gesture to him.
Gihun’s pain and suffering, but also all of his effort and the strength in his belief still being wrapped up in this jacket is what Inho had in mind, what made him feel close to Gihun’s spirit. He wanted Gayeong to feel as close to her dad as possible too, so he let go of this last piece he had left of Gihun to give it to her, as a final act of love that he could do for Gihun.
I know the opinions about this scene are controversial, but to me personally this is one of the most beautiful inhun moments because it perfectly captures both the twisted and yet still so deeply human feelings that Inho has for Gihun.
He was the one who put him through all this pain with the intention to break him, but in the end it was Gihun who broke him. Inho has to face his own guilt and what it means to feel human again, to have had someone that he now feels pain and grief for, after believing that he would never be able to feel these kind of emotions ever again.
I also love how warm and soft Inho looks in the scene where he gives the jacket to Gayeong. He’s being embraced by the light, it’s like he’s holding onto the same light that he saw in Gihun, the light that Gihun left behind for him.
Just how Gihun is embraced by the light that looks like a halo during their confrontation scene, which perfectly captures how Inho views him at that point. He sees all of his flaws and moments where he morally breaks, but instead of looking down at him he actually admires and envies him deeply. In my opinion we could even see a sort of relief on his face, when he saw that Gihun was incapable of fulfilling the offer that he gave him. (LBH also said this about the scene)