I think the exact boundaries to use for comparison are really difficult to pick, and vary on an individual level. For instance, the superb connectivity of the public transportation makes me feel as if Brooklyn, Queens, the Bronx (sorry Staten Island) are very much a part of NYC. I did not feel this way living in Mountain View; SF felt like a major chore to get to and a different world of its own. Plus it helps that the boroughs have the urban amenities I associate with a city that the suburban Bay Area towns like Mountain View lack.

So, while you might personally want to compare SF to Manhattan (which is totally fine based on what you value), when I think of my own quality of life metrics, I consider the NYC boroughs (excluding Staten Island) a cohesive unit that the SF Bay Area is not.