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

[Belacqua]

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.