Whether or not the instinct manifests to some degree in everyone, it truly goes into overdrive once you become a parent. It's hard to describe in all its scope. Even a
concept of a child being hurt by someone starts to become extremely disturbing.
For example, I've noticed that ever since I became a father, I can't enjoy some of the movies and shows I've liked previously, because the scenes where children are threatened or implied to be hurt become emotionally overwhelming to watch. To use a light example, take Star Trek TNG: Power Play (5x15). It was one of the episodes I always found boring, but when I got to it during the last semi-regular TNG rewatch, I had to pause it and collect myself. Almost switched it off. All because of one scene, where a mother and a child end up in the middle of a hostage situation.