"What do you mean our transit is bad, look, our ridership numbers are 3x higher than all our neighbors combined!" *Does not mention the fact that congestion pricing in neighboring cities is 3x lower.*
In the worst cases, it could even become a regressive tax of sorts. If your city has safe districts with good transit where rich people live, and unsafe districts with terrible transit where poor people live, congestion pricing will allow rich people to choose between the convenience of taking a car with no traffic jams versus the cheapness of transit, while forcing poor people to choose between a car they can't afford versus walking down a street where they may be assaulted.
It's even worse if you have rich people living in the city center where they work, and poor people who also work there living in towns much further away. Then, only the rich are able to vote on congestion pricing.
This probably doesn't apply to New York specifically (not an American, have never been), but it's definitely something to have in mind in general.