Souls on board is a way of asking number of living humans on board — so there’s no confusion when asked how many are alive on the flight whether we should be including dogs, cats, parakeets or the dead guy in a cargo casket being transported.
I feel like "souls" is not at all clear about the dogs :)
There's also the confusion between "passengers" vs "crew" vs "children" or "babies". Souls conveniently includes all of those groups and excludes irrelevant factors like dead bodies.