So: it's highly progressive when you consider all NYC residents, including those who don't drive. And it's maybe a little regressive between the wealthy and the very-wealthy, just on the general pattern that the very-wealthy spend less of their income. So, boo-hoo for the merely wealthy.
Congestion pricing doesn't seem nearly as regressive as gas taxes or sales taxes, narrowly focused as it is on just those driving in an especially wealthy and especially congested area.
And the parent post's idea that it'd hit the "blue collar" especially hard is, without evidence, just hand-waving. Their time is valuable too, and commute/on-the-job-travel time saved, from pushing lower-value travelers to non-road (or on-road-but-shared-vehicle) paths, could be worth more than the congestion fees.
Why would you say that? A rich person would not spend hours every day commuting though rush hour traffic, or spend most of their day inside their work vehicle. They would buy or rent a central home, possibly near the central office building where they work. City driving in itself makes for a very low quality of life and most rational people would minimize it, given the option.
So I would expect there's either no driving-income correlation, or it's inverse. So reducing congestion would make life easier for rich people, who don't care about the money and only care about the time - by pricing out people who are, by definition, poorer and can't afford to buy that convenience. You could say that the resulting post-tax distribution is progressive, the rich paying and driving more; but that's just a way of distorting the initial argument, that introducing a flat congestion tax is strongly recessive - to the point of completely pricing out poorer people.
There's expensive tolls, and limited & expensive parking, but excellent public transportation – the USA's most extensive and most-used subway system. NY state has the nation's highest (regressive!) gas taxes.
New York is far-and-away the US city with the lowest rate of car ownership:
https://en.wikipedia.org/wiki/List_of_U.S._cities_with_most_...
Manhattan is the most likely place for congestion pricing. But, though rare in the USA, many Manhattanites never learn to drive! And only 8% of Manhattanite workers commute by car to work:
https://www.nycedc.com/blog-entry/new-yorkers-and-their-cars
(I highly doubt that 8% is correlated with the poorest Manhattanites.)
Car-owning households in NYC are significantly richer than zero-car households:
http://blog.tstc.org/2017/04/21/car-free-new-york-city/
So, people who elect to drive in NYC (outside of driving-on-the-job) are richer than the New Yorkers who don't drive. Further, New Yorkers are richer than other Americans. Ergo, a fee that falls on people who choose to drive in NYC is unlikely to be "regressive", barring any actual evidence to the contrary.
I have no doubt that's true, as in most other places on the planet.
The thing is, you are making an absolute argument: those who drive in NY are richer than those who don't. I don't deny that. I'm making a relative argument: within the driving group, the poorer ones will be priced out. You say those are not poor, but "merely wealthy". Our grand-father said they are blue collar workers. You both lack data, and I suspect that we will see a wide, gentle slope, with most rich people driving, but also an important minority of the lower classes, depending on personal circumstances.
The thing is, does the exact composition of the driving public mater for the regressivity to be a political problem? Some will be priced out and they will scream bloody murder, because from their perspective they were sacrificed in favor of the rich. You could go out and say "boo hoo, you are merely rich", but that's a politically imprudent move - a thousand personal stories will pop up about those marginal blue collar workers no longer able to feed their families.
So you could be right that it is not regressive in a broad, social justice sense, but that's not the sense in which the problem will be sliced. Rather, the vast majority of the affected merely rich will invoke the principle of the matter, and in principle it's designed to price out the poorer drivers, there's no doubt about that.
An idea I'm toying with is to distribute "tradable driving coupons" to all people holding a drivers license. A flat tax would develop on the market, so the net effect is similar to congestion charging. But now the tax does not go to the city, but to the poorer residents who opt not to drive. Ideally, fiscal credits should be earned, so as to not reward those who don't pay any tax and thus have no contribution to the infrastructure. This beautifully solves the political problem and makes driving a matter of personal choice: you are now rewarded for preserving the public good.