The fine print is that owning a car is a prerequisite for living in the US outside a tiny handful of cities and metro areas (namely: New York, Chicago, Boston, DC, the Bay Area, and Portland) which have passable public transit, and that owning a car (even before gas) is fairly expensive, particularly when you're poor and forced to buy a car that is very prone to breakdowns. The tax is not a great idea because it harms those people needlessly. If the object is to discourage driving, or buying/operating wasteful, unnecessary, or ecologically (or infrastructurally) damaging vehicles, there are better ways of doing so directly. Or alternatively, of encouraging people broadly to switch to the kinds of cars you want your citizens driving.
Oh, and I forgot to mention that the "passable public transit" often exclusively services the more affluent parts of the metro area in question. The El doesn't go to South and Southwest Chicago, broadly speaking, and for those parts which it does go to, even the residents of those regions advise against using it. BART doesn't go to East Berkeley, many parts of East Oakland and the extreme West of Oakland. Both it and Caltrain don't go to West SF. This is of course ignoring that SF itself is too expensive for most people under 6-figure incomes to live in, and so people are moving further East into Richmond, Concord, Dublin, Pittsburgh, Antioch, etc. which don't fall into the "passable public transit" regions.
When I lived in Santa Cruz, there was a day where I had to go to the DMV to get a copy of my driving record for a job. The nearest DMV was the next town over. Neither I, nor anybody else I knew owned a car, so I took the bus (only public transit available). Ordinarily, it'd be a 30 minute drive round-trip. It took me over 2 hours to make it there and back by bus. Cars are a necessity.