There is a big dodge in the debate about income "inequality" and wealth "inequality".
There are many things distributed unequally, like respect, that most people could care less about. The reason is that what they're really interested in is fungibility of valuable stuff.
If it isn't both fungible and valuable, no one worries about its inequality. If it's both fungible and valuable, they do.
So, we don't care about inequality of respect ... not because it isn't valuable, but because it isn't fungible. For example, someone could (in the spirit of the TV season) transfer their major award to you, but you wouldn't really feel like it's yours. Or, as a professor, I get more respect than many other professionals, but no one would be interested in me transferring some of that respect to them; it would be silly, and we all know it.
More to the point, many jobs are paid with in-kind benefits (like the employee of the rock/gem shop I was in on Christmas Eve who can buy the earrings at cost rather than the retail price). The distribution of semi-precious stones is seriously unequal, but no one worries about that because some people just don't find them that valuable. Alternatively, employees of restaurants can often buy entrees at cost, so again there's an unequal distribution of Big Macs and shrimp scampi. But we don't worry about that inequality because those items, while fungible, depreciate quickly enough that the fungibility doesn't matter much.
So no, we worry about cash income inequality: because the cash is valuable, and it's fungible. Once it's out of their hands, and into ours, we're richer and no one can tell why.
I'd go further and argue that this is why we're more concerned about income inequality than wealth inequality too (in spite of the fact that wealth is distributed far more unequally). Income is almost always in money. Wealth is often converted to not so liquid stuff ... that no one is interested in transferring.
In my case, as I approach 50, I have some of my wealth stored in phones-that-worked-the-last-time-I-used-them, open spice containers, polaroids, opened cans of paint, pens I've picked up somewhere, and so on. No one who's young, and most people who are poor, has such things. But they don't want them either. Instead, they want the portion of my wealth that retains value, and that if put in their hands, would give them value with no strings attached: mostly, that's money.
N.B. Most of this appeared as a comment on a follow-up post at EconLog to the post linked above.