It's strange, because NYC is simultaneously overpopulated with cars, but is also one of the only places in the US where you can reasonably live without a car (I don't have one).
I’d reckon it’s less than 20%, maybe even as low as 5%. The subway alone has a daily ridership of about 2 million, plus the massive amount of people who just walk or bike to the store, work, etc. I think it seems like more people are using cars than actually are due to their visibility, like you mention—they take up more space, they’re given more space, etc. It’s insane.