This custom is not confined to the Bay Area, but it does vary from place to place. It's not a function of American English, it's about the culture and governance structure of the area in question.

Someone from Brooklyn, New York might describe their home locality as "Brooklyn" or "New York City" (or indeed their neighbourhood in Brooklyn, e.g, Williamsburg). But it's perfectly valid for a person from Brooklyn or Manhattan or Queens to say "I live in New York City" as the five boroughs are all contained within the municipality of NYC.

In Los Angeles, someone from Santa Monica or Beverly Hills might just say to a foreigner "I live in Los Angeles" for simplicity, even though they're separate municipalities from the City of Los Angeles, though they're all within Los Angeles County. Whereas someone in Pasadena or Malibu would more likely name that as their home town, even though they're also in Los Angeles County.

I'm not so familiar with specific customs in many other major metropolises, but certainly the metro areas of Seattle and Boston are collections of separate cities - I.e., Redmond, Washington and Cambridge, Massachusetts are referred to by name as distinct from Seattle and Boston respectively.

But I'll again remind you of the whole point of this discussion: in Australia we have a fairly straightforward understanding of what is meant by city names like Melbourne or Sydney for the purpose of discussing a topic like this news story: it refers to the greater metropolitan area.

In the US it is less straightforward and sometimes quite different depending on where you're talking about, e.g., SF Bay Area, hence the root commenter's helpful clarification for US readers.