Harry Potter's favorite Peanuts character is probably Linus…and not because Linus is an intellectual who drops startling deep wisdom on Charlie Brown & Co.
Harry has one sentimental artifact from his childhood: the blanket he was wrapped in when his parents died. (Whoa, that's morbid.) Harry received it from Dudley (good ol' Dudders) after Aunt Petunia died.
Anyway, Harry loves his blanky, giving it to Albus because,
HARRY: This… is the last thing I had from my mum. (1.7.20)
It would be sweet, except it isn't. Harry is still treating Albus like his own personal mini-me. That might work for James, but Albus isn't James. Albus isn't Harry. Albus is Albus, and he doesn't share the same sentimental attachment to his dead grandparents that Harry does.
However, Albus later uses the blanket to deliver a message from the past. By doing so, he effectively ruins the blanket—burning an SOS message into it. But Harry doesn't care about the destruction of his blanket. He cares about his son more, and he never once gets upset about his sentimental item being ruined. Maybe he's finally put his past behind him.