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I've heard SF described as a Brooklyn without a Manhattan.

Which generally checks out in numbers. Brooklyn's density is ~14,700/km^2, SF's is ~18,500/km^2.

By comparison, Manhattan is 27,800/km^2.

You mixed up some units here: Brooklyn is way denser than SF (about twice as dense). Your SF number is people per square mile, not per square km.

Same units, Brooklyn is ~35,300/m^2, SF ~18,500/m^2

Manhattan ~69,400/m^2

My figures are based on square kilometres. A square mile is quite different.

Edit: bonus round! I got Manhattan right but did the reverse error for SF (quoting km^2 for miles^2).

Where did you get 18,500/km^2 for San Francisco? 870,887 / 121.46 km^2 = 7170. More like Queens, not Brooklyn.

Ah, I got it wrong. Mixed miles and square kilometres.