In short the present tense "I am the God of Abraham" does not seem to necessitate the idea that Abraham is still alive.
It depends on what "god" means here. If a god is truly all powerful, then why should death matter at all? Can't Abraham be resurrected by God (if God wishes)? On the other hand, if the meaning is something more like "commander" rather than "creator", I can see the distinction. But I don't see how that is (narrowly) a linguistic issue.
Historian: "I wish I knew who the owner of the house was so I could interview them about the "Great Burning."
Bystander: "I am the owner of that House"
Actually, that is problematic in two ways:
1. We usually would say "was" instead of "is" in that case. It's more natural. I'm not sure that "is" must be considered 'wrong', but it's not what I would say.
2. The definite article suggests uniqueness, and a house may have at different times belonged to many owners (who sold it to the next, etc.), and tense should line up with that. To say "I am the last owner of that house" would be perfectly fine, I think (and maybe even odd with "was"). But to say "I am [still] the owner of that house" sounds odd (and better with "was", without "still").
This reminds me of how when I was in a small town in Costa Rica there were no street addresses and instead all directions were given based on landmarks. I stayed at a place on the hill near the water tank, so directions were given relative to that. But a very central part of town was where there used to be a very large, old tree. It had fallen over years before and was gone now. But everyone knew where it was. So directions were still along the lines of "turn left at the big tree", or "across the street from the big tree", etc. Obviously that seemed very strange to me (even once someone had told me there used to be a big tree there!), but the locals didn't seem bothered by it.
There are some interesting distinctions regarding whether an entity is still alive in semantics. One argument is regarding the English present perfect. Compare these sentences:
"Einstein has been one of the most influential physicists."
"Einstein was one of the most influential physicists."
"Einstein has taught at Princeton."
"Einstein taught at Princeton."
Unless you can establish some present relevance (for the first sentence it might be easier), the present perfect sounds odd.
But of course Einstein is not a god. And those examples are about subjects, not objects.
One suggestion: really look into the original Greek (or Hebrew if it comes from something earlier) to see what's going on there. No idea if this translates well into English.
