I used to really dislike this name, but lately Sydney (for a girl) has grown on me. It does feel a little 1980s (despite its popularity actually peaking in the early 2000s), but I don't think it's too dated. It's in the same category of names as Lindsay / Lindsey, Leslie / Lesley, and Ashley; but Sydney is my favorite.
Since I am not Australian, and have never been to Australia, I can't comment about the city's name.
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