(not to sound sacrilegious, but) I think the best analogy for the Trinity would be something like "Cerberus the 3-headed dog" (Greek Mythology).
Purely for the sake of analogy...
We may suppose that Cerberus has three brains and therefore three distinct states of consciousness of whatever it is like to be a dog. Therefore, Cerberus, while a sentient being, does not have a unified consciousness. He has three consciousnesses. We could even assign proper names to each of them: Rover, Bowser, and Spike. These centers of consciousness are entirely discrete and might well come into conflict with one another. Still, in order for Cerberus to be biologically viable, not to mention in order to function effectively as a guard dog, there must be a considerable degree of cooperation among Rover, Bowser, and Spike. Despite the diversity of his mental states, Cerberus is clearly one dog. He is a single biological organism exemplifying a canine nature. Rover, Bowser, and Spike may be said to be canine, too, though they are not three dogs, but parts of the one dog Cerberus.