Price as a dad, who isn't exactly abusive but definitely absent, both physically (deployments) and very distant emotionally, not a great father, struggles with dropping the strong, authoritative persona he has to be while on duty when he comes home. His son grows up hearing 'no, don't bother your dad right now', 'he has a very stressful job, you have to understand', 'don't say anything, don't antagonise him' from his mother, especially when Price drinks, learning to walk on eggshells when his dad is home. His son becoming so much like him as he gets older (whether he wants to or not), taking up space in the same way, the restlessness and fidgeting, mimicking his tone and repeating certain phrases (probably without even realising), so they end up sounding almost identical sometimes. And for John it's like looking into a mirror when his son clearly struggles with his temper, volatile, clenching his fists and trying not to snap at the smallest thing. Price, who knows that the things he hates most about himself have his father's blood all over them, wondering if his son feels the same way about him. Price sees in his son his past, his son sees in Price his future. They both know it, neither one of them will mention it.
And I can't see his son having a neutral response to the similarities between them. He'd either want to be exactly like him, actively trying to follow in his footsteps, copy as much of his father as he can (he will never get the approval he craves) or nothing like him, trying to become the exact opposite (it won't work, Price felt the same way about his father and will now watch his son unintentionally do the very thing he was trying to avoid)