[Serious] Could an omnipotent and omniscient god prove that he was God?

[Objectivist]

To exist is to be necessary

It may be that you and Neo are coming at this with different definitions, which would complicate things. 

Generally, in metaphysics, existence is not always necessary existence. There is also contingent existence. 

This is the first thing that comes up when you Google "necessary vs. contingent":

A contingent truth is one that is true, but could have been false. A necessary truth is one that must be true; a contingent truth is one that is true as it happens, or as things are, but that did not have to be true.

Pretty much the whole of classical theology relies on the difference between things that exist necessarily, and things that exist contingent on other things. Most things are in the latter category. 

Though you may be using a definition I'm not familiar with.

Yes, Belaqua, I understand this, but that something is generally accepted does not make it true.  I explained where this whole necessary-contingent dichotomy comes from.  I understand it down to its roots and it is based on a fundamental error.  The error is a faulty understanding of concepts and how they are formed and defined.  The error is, among other things, the result of thinking that a concept's meaning is its definition.  Those attributes included in the definition are considered necessary and those not included are contingent, we could imagine them being different without contradiction.  Also known as the analytic-synthetic dichotomy or the rationalism-empiricism dichotomy.  It is a false dichotomy.  There are not two kinds of truth, one that can be known apriori by analyzing the definitions of words and the other synthetic, based on empirical observations.  According to this theory, analytic truths are not factual.  They have no relation to facts.  Synthetic truths are not certain since we can imagine them being different.  This is something you should study and learn because this false dichotomy is everywhere.  It has wreaked so much havoc in the world.  

A concept's meaning is not its definition. A concept's meaning is the concretes that it subsumes and all of their attributes both known and unknown. This error of supposing that concepts mean their definition strips them of all objective meaning and makes them 'social constructs'. This is why you see a lot of definitions of gods containing the word necessary. Notice the reliance on the primacy of consciousness inherent in this false dichotomy.

I follow the objective theory of knowledge, one in which knowledge is not acquired apriori but is acquired by means of perceiving reality and identifying what is perceived by an objective means.  In other words, logic applied to observed facts.  There is no reason, in reason, to split truth in the way the analytic-synthetic does. 

So if you accept this theory of knowledge you have to say that God is an analytic truth because gods are defined as necessary beings apriori and they are not related to facts in any way. They can not be proved by looking at facts but only by manipulating words which have no objective meaning.