Halcon: But to answer your question, that person would very likely tell you which God they feel in their heart and why. Or maybe they couldn't.
That last part speaks to the point I am making. The person who tells me that he experiences god in his heart can claim that it is one god or another, and there is no way to challenge his claim that everyone can agree upon. Indeed, his description of "god in his heart" may not match anyone else's. But I bet his description of heartbreak, or of what he felt when he touched a hot surface, would be easily and accurately understood by everyone else.
Keep in mind that I am accepting your approach at face value, and even that only gets us as far as "maybe it could be." Gods exist in a muddy world of conjecture, in a place that we define however we wish. And that is the extent of our evidence for them. For a concept of such tremendous potential importance, that is a shockingly thin thread to depend upon.