I think everyone who lives near a major city calls that city "The City" in everyday conversation. I live near NYC, and I do it in reference to Manhattan. (Less so the other parts of NYC, such as Queens and the Bronx.)

I've only heard it happens around SF and NYC. Certainly nobody in Redmond was calling Seattle "the city" when I was there.

Other commenters have pointed out that almost every major metro area gets called "the city" by its suburbanites, and that matches my observations of at least Seattle, DC, and Nashville. I've lived outside Seattle for 5.5 years and routinely hear it called "the city" by both locals and transplants.

I grew up outside of DC, and people would say it there as well.

In Santa Cruz we typically called it "SF", or else the full "San Francisco". Didn't hear anybody use "San Fran" or "Frisco", and "The City" was too nonspecific.