It's very easy to make the logical leap that increasing inequality is equivalent to "the rich get richer and the poor get poorer". And most people don't necessarily have a problem with "the rich get richer" part, but they do object to "the poor get poorer".
It is a leap though. Bezos, Gates, Musk, Buffet, Page, Brin, Zuckerberg all started as regular people and became ultra-rich through their work and ideas.
But it's not a complete fallacy either. If you are a real estate owner in New York City, your assets continue appreciating at a good pace, well above inflation, and you have a vested interest in NIMBY, which does indeed restrict social mobility.
Social mobility, or social mobility opportunity are harder to measure though, so a lot of people are ok with using the inequality as a proxy. I do agree with you that it's an imperfect proxy, and focusing on it can have negative side-effects.