I have two theories on how the same "god" can be so many miles apart. One is that the Mormons claim that ancient Israelites came to the Americas. As the Bible states, the Israelites were known to turn away from Yahweh god and adopt other religious practices, ie the golden calf and also possible Babylonian gods. So maybe they brought that with them. But I haven't read the book of Mormon to get the whole story.

The other idea relates to Peleg, also in the Bible, "during the time of Peleg that the earth was divided". Perhaps Pangea was the original map of humanity, then after The Tower of Babel situation, the continents were split apart. It seems plausible to me when you consider that the same structures are found globally, like the pyramids, and figures like this guy. Also the Olmec statues looking like Africans. Africans aren't known for seafaring. I also find it hard to believe the Pacific Islanders travelled thousands of miles to get to their eventual homelands by small Catamaran type boats. Just watch videos to see how rough trans-oceanic travel can get. Those waves are insane, the boats would have capsized. I doubt that Israelites or Islanders were able to make those journeys. But a massive cataclysm of moving continents could account for that. Or lizards floating on rafts of palm fronds as biologists would have us believe.