Is chloride necessary for animals?
My answer may be a little basic, but I think it is comprehensive:
Many elements are special; for instance, many metal ions such as iron or cobalt are required as cofactors or in catalytic sites due to their chemistry. Oxygen is highly electronegative, that also grants it some of its functions. But... without delving too much into things... chloride is the most basic constituent of saline, without which it would not be saline. Here are 3 very, very important and yet

It works well with alkali and alkaline earth metals, hydrogen, and many compounds... it's a natural pairing!
You cannot separate positive and negative charges in a solution at a macroscopic level. It's not a feasible thing to avoid harboring chlorine atoms as part of biochemistry.
It accounts for a large portion of extracellular fluid tonicity.
etc...

Thus, most inter- and extra-cellular fluid is relatively chloride ion rich, across the entire bush of life. To my knowledge there is no life, tissue or cell on this planet that does not contain chloride ions as part of its makeup by necessity. It is simply heavily constrained in evolution as a basic component. It's just not practical to obtain K+ or Na+ or Ca2+ ions without Cl-, it is so common in nature.
If you want specific examples, you can always mention its indispensable roles in neuron firing (transmission of axon potentials) and general cellular homeostasis, which is dependent on electrochemistry and osmolarity and stuff like that which is heavily influenced and dependent on the presence of a regulated concentration of solvated Cl-.