It's not called "Bridge"—it's the Sydney Harbour Bridge. The name is descriptive, not arbitrary.
Together: "The bridge across Sydney Harbour." Similar to "Golden Gate Bridge" (San Francisco) or "Tower Bridge" (London).
Sydney has multiple bridges: Anzac Bridge, Gladesville Bridge, Ryde Bridge, Captain Cook Bridge. "Harbour Bridge" distinguishes the iconic harbour crossing.
The official name has never been just "Bridge." It was the Sydney Harbour Bridge from the 1922 enabling act through construction (1923–1932) to today.