I am skeptical there is a sizable amount of low-income people that were driving into lower Manhattan on a daily basis. You cannot find daily parking in this area of Manhattan for less than $30 either way.
There are also exemptions for people who make less than $50,000 a year
It's not as progressive of a tax as a progressive income tax, but it is definitely a tax which will help coordinate behavior and will disproportionately be levied on the wealthy. If any policy that wasn't perfectly progressive could not be implemented, we couldn't have car registration fees, subway tolls, or sales taxes either.
I live in Brooklyn and there are times I want or need to leave the city to Jersey or PA. I'm not trying to park or even be in Manhattan but adding an extra hour or two to a trip to avoid Manhattan is costly. It already costs me close to $30 to leave and come back if I head west.
Also, I live in a low-income neighborhood 1 mile from Manhattan. Many people have cars and use them for work and family. Transit can really suck the further out you are.
I'm pro-transit and am not a giant fan of cars. I use mine for transporting my work as well as some trips. Using Uber would maybe save me money if it was only me travelling short distances and the driver doesn't mind me loading paintings or sculptures in their car or SUV.
I don't even know if I disagree with the law but it's not going to stop the less wealthy from bearing the brunt of this. To act as if driving into Manhattan is purely an act of entertainment or easy choice is missing a million elements. What if your older disabled relative lives in Manhattan and can't take transit? Uber will cost more than the $15 even one way. What if you're unable to take transit and just want to visit someone in Manhattan? This is not a tax on the rich as the rich are maybe 1 in 10.
Pricing lower-income folks people out of an activity entirely doesn't make it a "tax on the rich".