Does the concept of the Triune God violate the Law of Identity and the Law of Non-Contradiction?

The Trinity doctrine is  3 individuals that share the same essence. So how do you prove an essence and how would you define it? One definition is : "something that exists : ENTITY" another : "The individual, real, or ultimate nature of a thing especially as opposed to its existence." I do not think logic is outside of theology but I am not sure their is anyway to prove this. Their is nothing in the natural world to compare it too and the only comparison I would think would be a 3 headed person. Maybe violate the law of  identity but I do  not think the law of non-contradictions.

You present this: “The big question is : does each of these persons have a unique identity?”

This, I think, places you too high in the hierarchy of the logical process. It would help if you first defined the gods of which you speak. You cannot assume anyone understands what your definition is until you tell them. 

Logically, in any discussion, one must present (1) an intelligible definition of the subject under discussion [in this case god] and (2) adduce evidence to support the proposition [in this case, the existence of the god defined]. No one has ever presented me with (1), so we never got to (2). 

To me, the problem faced by those who profess a belief in a god — any god — is not they cannot adduce any evidence to support their belief. The problem is first to define what the god is, then discuss other aspects of that god. 

Before we can discuss whether the Trinity violates any laws of logic, we need to be correct about what Trinitarianism is. 

it is not Harrison Ford  playing different characters at different times. This is another christian philosophy called modalism.

A more appropriate analogy would be the various actors who have played James Bond. Pierce Brosnan is a unique person, but is also James Bond; Sean Connery is a unique person and is also James Bond; Daniel Craig is a unique person and is also James Bond. Daniel Craig is not Sean Connery, Sean Connery is not Pierce Brosnan, and vice versa. However, they are all James Bond.

Likewise, Trinitarianism states there are three distinct persons that comprise the Godhead. Imagine a monarchy ruled by three kings. They are each equally God in that they all share the same "hypostasis", or "essence"

While its possible that specific beliefs about the Trinity might violate laws of logic, the general concept doesn't seem to violate the principles you mentioned

Theology has its own logic and many people have asked the same question. BTW, your line of argument is what originally gave rise to Unitarianism in the 18th Century. 

The basic idea is that God has multiple identities with a single will.  So we have a number of separate 'processes' that all lead to the same 'conclusion'. 

It does not violate logic any more than 2+2=4 producing the same result as 8/2=4 and 8-4=4 does.  The single outcome (one will) is arrived at by the multiple separate persons. 

Well, yes, in the Aristotelian sense, it clearly does. This is not to say that theologians, philosophers, and religious apologists (from time immemorial) have not spilled gallons of ink trying to reconcile with it.

Firstly, consider the historical reality that "the concept of the triune God" is one of the earliest constructs of the human imagination.

“Many who believe in the Trinity are surprised, perhaps shocked, to learn that the idea of divine beings existing as trinities or triads long predated Christianity. Yet, as we will see, the evidence is abundantly documented.” Marie Sinclair, Countess of Caithness, Old Truths in a New Light.

“It is generally, although erroneously, supposed that the doctrine of the Trinity is of Christian origin. Nearly every nation of antiquity possessed a similar doctrine. All the ancient nations believed in the Trinity ” The early Catholic theologian, St. Jerome p. 382).

Even Aristotle wrote: ‘All things are three, and thrice is all: and let us use this number in the worship of the gods; for, as the Pythagoreans say, everything and all things are bounded by threes, for the end, the middle and the beginning have this number in everything, and these compose the number of the Trinity’ ” (Arthur Weigall, Paganism in Our Christianity, 1928, pp. 197-198).

The typical theological strategy is either to spin The Laws or Move the Goal Post altogether. Mostly this consists of embracing the Laws of Identity and Non-Contradiction but move them beyond human conceptualization and place them immutably in the 'mind' of God. (See Ronald Nash, Gordon Clarke, Cathcart and Klein, etc. ) 

The Hindus had an especially novel approach to your Harrison Ford analogy by depicting its triune Gods Vishnu, Bramha, and Shiva and explaining there is but one God, Brahman, in its three manifestations (lilas) as creator, preserver, and destroyer. They even invented their own Law of Acintya Beda Beda Tatwa or simultaneous One and Difference.

While it can be an endlessly fascinating and enlightening line of research and discovery, it's hard not to resolve with Thomas Jefferson's delightful rebuke of the whole confabulation:

"Ridicule is the only weapon which can be used against unintelligible propositions. Ideas must be distinct before reason can act upon them, and no man ever had a distinct idea of the Trinity. It is the mere Abracadabra of the mountebanks calling themselves the priests of Jesus.” Thomas Jefferson Letter to Francis Adrian Van der Kemp (30 July 1816), denouncing the doctrine of the Trinity - 1810 )

Or, as the philosopher, Andrew Bernstein writes, “The tragedy of theology in its distilled essence: The employment of high-powered human intellect, of genius, of profoundly rigorous logical deduction—studying nothing.”