Redistributive effects of a carbon tax are more of a problem (particularly for liberals) than what’s implied here.
One of the points of the article was the the average income for people will increase over time, increasing the economic means to fight climate change. That is saying the the economic means of society as a whole increases. It doesn’t necessarily mean that the economic means of a poor or median individual improves.
Carbon intensity is also a problem because wealthy people have fewer emissions per dollar of output. If a government imposes a flat carbon tax with no redistribution, it is a highly regressive tax right out of the gate. In order for this to help redistribute wealth, use of the proceeds of the tax must be more progressive than what the tax is regressive. This might be easy on a national scale (if you had a relatively leftist government to begin with), but much more difficult on an international scale.
The most likely interplay between inequality and climate change is that inequality presents an additional political barrier to instituting climate change action. Firstly, it’s in the interests of the rich to resist new redistributive taxes, and secondly, there are numerous ways to shape those taxes in a way in which it will increase inequality.