An axiom is a true self-evident proposition.
A proposition is considered 'self-evident' if in the act of refuting it, you confirm it and it is impossible to do otherwise.
This can be done empirically ('Existence Exists' is an axiom because to deny that existence exists would mean deny existence including the existence of the phrase 'existence doesn't exist' which you can empirically verify as having existence) or it can be done logically ('There is such thing as Truth' is an axiom because to deny this is to posit that 'There is no such thing as Truth' is true, which is not logically justifiable because either there is no such thing as truth and the statement 'There is no such thing as truth' is not true or there is such thing as truth in which case the statement 'There is no such thing as truth' is false.) There is no way to deny these axioms without confirming them.
Axioms tend to work off of each other (for 'There is such a thing as Truth' to be true 'Existence Exists' must be shown to be true first) and thus Metaphysical axioms generally lead to Epistemological axioms which is where axioms tend to 'die off' as only so much of our experience and therefore the truth can be know axiomatically (due to the problem of our senses, which can be fooled.) Epistemological methods are then used from there to deduce and infer truths among the other various branches of philosophy.
The 'mathematical' axiom your friend is talking about is actually a different class of axioms (and defined differently) from a philosophical axiom which one might used to prove that God exists. Mathematical axioms are abstract representations of the imperfect shapes and dimensions of the things we find around us. Much like God, a 'line' as Mathematicians defines it, doesn't exist anywhere. It is a semi-arbitrary figure (based on what appear to be lines around us) which is used for practical purposes. In fact, much of the problem of Quantum Mechanics is due to the fact that the measurements that we have are imperfect measurements of what actually exists (though not by much) and we actually can't get a perfect measurement. They are 'axiomatic' in the sense that they are assumptions which mathematicians use as a 'ledge' so to speak from the world of experience to the world of abstractions which math and geometry deals with and back again. Because it's impossible (or at the very least we haven't been able to do it) to translate our exact experience into abstract measurement, mathematicians accept these assumptions as, essentially, 'close enough.'
God is not axiomatic, ultimately, because you can falsify God without confirming His existence.