Its a fallacy to call it a fallacy though. Sure, many forms of wealth is not zero sum, but some are. For example political influence correlate with wealth and is basically zero-sum. So larger economic inequality concentrate power among fewer people. This is pretty significant whether you consider it a good or a bad thing.
There's no doubt it's in our DNA. Forming groups with likeminded people is what kept our species alive for hundreds of thousands of years. And it's only been in the past ~150 years that we as a civilization have started to seriously care about the wellbeing of others unlike themselves. We've undoubtedly come a far way, but that's not to say we don't have much further to go. Fighting with our innate tribalistic tendencies is hard.