Primary nickname: "City of the Golden Gate" — from the Golden Gate Strait (not the bridge). The strait, named by John C. Frémont (1846), is the entrance to San Francisco Bay from the Pacific. Chrysopylae = "Golden Gate" in Greek.
The Golden Gate Bridge (1937) borrowed the name—then *became* the icon. So the city is named for the strait, famous for the bridge.
Call it San Francisco. Or The City (if you're from the Bay). The Golden Gate is the strait; the bridge is the postcard; Baghdad by the Bay is the poet's wink.