A fireman doesn’t go around posting photos of the baby he rescues using them to get a raise or in a campaign to win a political office, either.
Are you comparing a baby to a dog? Rescuing baby means the baby is returned to his or her guardian/parents. Rescuing a dog is like taking home a used car... there are some procedures you have to follow, but you can (try to) return it or sell/give it away. Dogs have no rights except to not be tortured. They do not have the right to live, but if they must die, they should not needlessly suffer.