For instance, it assumes transportation costs in Manhattan are higher than most other places. This is unbelievably inaccurate in my experience. Maybe it assumes people here take taxis all over the place? I've found transportation to be cheaper in NYC than anywhere else in the USA. No car or gasoline or insurance or maintenance, etc. A car costs on average about 9k/year to own according to AAA. In a place like NYC if you take the Subway every day your cost is about 1,300/year. Throw in a few hundred for cabs and you're looking at 1,800/year. It doesn't cover the rent gap compared to most places but it helps.
In effect this calculator says a salary in Chicago goes nearly twice as far (so if you make 200k in NYC that would be 106k in Chicago. In my experience that is simply untrue. 200k in NYC would be closer to 170k in Chicago.
I think this calculator is making the assumption someone would require the same resources, like a car in all places. Or that someone who had 1600 square feet of living space would expect the same in NYC, which would be ridiculous. No reasonable person would assume they would move to NYC and get a car and garage it. That simply isn't the typical lifestyle here. Most my friends born and raised in the city don't even have licenses.
It also claims that living in San Francisco is significantly cheaper than NYC which is untrue as well.
For example, it's not exactly sound to average the price of all neighborhoods in Manhattan.
I hope it would be apparent you should reference many different sources when comparing salaries in different locations.