There’s eight.
Eveyone always forgets the Antioch bridge.
ignoring pairings of bridges, we’ve got, working roughly north to south:
Antioch
Carquinez
Benica-Martinez
Richmond
Golden Gate
Bay
San Mateo
Dumbarton
To beat a dead bridge, the KCBS traffic reports aways call it the Bay Bridge, as in “The Bay Bridge is backed up into the Maze.” On the other hand, I’ve always heard the other bridge called the Richmond-San Rafael Bridge, but the San Mateo bridge is never the San Mateo-Hayward Bridge.
Me, I take 237 and avoid the tolls. 
Yeah, but who wants to go to Antioch? 
Seriously though, that’s pretty much past the Bay and into the Delta.
During the years I worked in San Rafael but lived in Berkeley, I learned that the north bay bridge has two names, always referring to your destination. Folks in Marin call it the Richmond Bridge, and folks in the East Bay call it the San Rafael Bridge.
As Voyager points out, the same does not hold for the mid-southerly bay bridge. It’s always the San Mateo Bridge, whichever side you start out on.
And although the official moniker “San Francisco-Oakland Bay Bridge” is still seen and heard from time to time, as has been firmly attested to in real life it’s the Bay Bridge.
Perhaps, but Caltrans says it’s in the Bay. It’s their bridge, so I gotta go by their decision.
As for going to Antioch? Ugh, I have no idea. What amazed me is to find that Antioch actually has hotels. A friend is getting married this weekend, and their father is flying in from somewhere East. The flight is coming into Oakland, and Travelocity booked them into what must be the cheapest hotel in the whole Bay Area, and amazingly, it’s in Antioch.
As a resident of the town next to Hayward, on-and-off for much of my 50+ years, Hayward is a place few of us care to promote. Not only is there “no there, there,” but there is little reason to put any there there.