I imagine that Error has some sort of parental instincts when it comes to Goth that never really left, even when he stopped being Geno. Those parental instincts transferred when he started taking care of Paperjam, but stayed consistent with Goth. For a moment in time, people thought Error had a soft spot for children, which was almost immediately proven incorrect after he almost punted Pallet like a football for whatever he deemed a criminal offence at the time.
I imagine a situation where goth is put in actual danger, whether it be harm from another Sans or a different type of harm, causing Error to go ballistic. At the end of the day, Error is an emotional mess of a person, who switches from nonchalant to full of Rage at the snap of the finger, to see him fueled with anger for something that is inherently a part of him is not too surprising. He doesn't inherently know why this feeling brews so strongly and why it pulses through his soul so deeply, but all he knows is that iit s his kid, and may the Multiverse collapse before he lets something happen to his kid.