Living expenses vary in a lot of complex ways, yeah, depending on your lifestyle. I find cost of living here overall cheaper than in the SF Bay Area for my own lifestyle though, mostly due to transportation and healthcare. I was able to sell my car, ditch my gasoline/insurance expenses, and no longer have to pay co-pays or employee contributions for my health insurance. People with other lifestyles may find it more expensive, especially if you want to buy a car (which has its own separate, very high taxes).

I'm not sure VAT is a big component of the difference. I think housing costs and high wages are the biggest factor. Grocery prices are slightly higher (perhaps due to VAT), but we're talking differences there that add up to maybe 1% of my income annually. Eating out is much more expensive, mainly because everyone employed in the restaurant is making at least a lower-middle-class salary (~$40k or so... nobody's working for $7/hr). Rental housing in Copenhagen is more expensive than most of the Bay Area, but cheaper than SF proper or NYC. Housing to purchase is actually quite cheap; you can get a 2bd in reasonably central Copenhagen for $200k, which is completely impossible in SF or NYC.

I suppose the PPP comparison is made against the U.S. as a whole, in which case cost of living is definitely higher than, say, the midwest or Texas. But I don't think it's particularly high compared to coastal US prices. If you want to buy housing, if anything CoL is considerably lower.