Why are the hormones dopamine, adrenaline, noradrenaline and serotonin so ubiquitous across the animal kingdom?
Evolution.

This is not special to hormones: receptors and ligands of all types are highly conserved across not only the animal kingdom, but also to single-celled organisms. The degree of conservation is typically dependent on the closeness of evolutionary relationship: more related organisms share more homology.

Because species evolve from common ancestors, they tend to share a lot of the same 'tools.' Changing the purpose of a hormone would likely require some sort of selective pressure, some new reason to use that molecule in a different way, and if the old and new roles conflict then there needs to be a replacement. This can happen, but it's complex and requires just the right conditions. More often, minor changes occur: among the steroid hormones, for example, you will find many small biochemical tweaks that make new molecules that have different functions.

Certainly the functions of some hormones are not entirely conserved, and if you look at the nervous systems of, for example, insects versus mammals, you will find some major differences, but the closer in evolutionary time you look the more similarity you will find.