Sam sees Dean in their Dad's jacket as an extension of the familial divide in season 1. He sees dean picking a side and wearing the uniform of their father's army. They were raised like soldiers and even as they're supposed to be set off alone Dean holds more strongly to John's teachings and Sam can see it every time he watches Dean shrug on that jacket. Like he's taunting him with it.
Dean taking the jacket, even as a kid, is something Sam could never get away with. (see: john pulling a gun on a 18 year old sam when he got upset at john taking his hotdog. im not kidding) If Sam had taken the jacket, it would not have gone unmentioned, it would not be addressed with a mere look. Sam is punished for simply reminding John he exists, for taking up space and advocating for his emotional and physical needs all throughout childhood. Taking the jacket would be further encroaching on John's adamant ignorance of the child with something ever so slightly wrong with him that he can't quite look at. Taking the jacket would force John to look too closely. And we can't have that.
But the jacket is grief, the jacket is fatherly comfort taken by force. And so Sam doesn't get that. Doesn't grow up feeling safe around John, isn't allowed to grieve or acknowledge or feel his mother's death. He sees how he is excluded and grows resentful, more vocal about other areas he isn't supported. Goes on about moving around too much, because he can't shrug into the fantom hug of his father for support as he moves through school after school, drifting. Not even tethered to John, not even tethered to the grief that defines his family so deeply.
So yeah. He doesn't like the reminder that is Dean wearing the jacket.