Religion and the supernatural are often closely related, but are not tied. A religion can be the worship of a supernatural being, the worship of a 'thing/object' or a following to a principle or certain "way of life". While most religions are tied to occult things, not all of them are.
Your girlfriend on the other hand, depending on her "god", does indeed believe and accept the supernatural as existing. She may have certain biased as to some unnatural occurrences, but she obviously doesn't deny them all.
I bring up what is God then?
Quite easily explained as a
Supernatural life form, or false title.
She brings up what is the supernatural?
Something that exists outside of the laws of nature. No, her god does not fit in with nature, on any level, unless she worships nature as her god.
I point out that the gods of Norse and the Greek have both fallen, and they are "mythology," while Christians are "religion."
You seem to have a relatively fine grip to the idea of personal gods being fallible, but you still believe in the supernatural? Why?
So I wanted to bring up are they similar?
Somewhat.
Are they both figments of imaginations and will they all die off and get written off as mythology in another 1000 years?
Why should the religions of our generation be so special? More directly, why should YOUR religion be so special? Thousands of faiths have come and past, it's funny how quickly a Christian can point out the obvious flaws, and falsehood of Hinduism, yet they can't see it in their own. Ignorance is bliss, no?