As dmurray commented there's differences in what is counted. Another difference is that Paris proper has relatively small parks, whereas Berlin has several huge ones as well as lakes.
So yeah, Paris is dense, but if you compare block by block the differences are smaller.
Even the (by far) most dense borough of Berlin is only at 14,373/km²[1], though. That's considerably lower than almost all of the Parisian boroughs[2]. Even the whole city's average is 20,000/km². From the point of view of most American cities with some obvious exceptions, the difference is probably indeed not that noticeable though.
So yeah, Paris is dense, but if you compare block by block the differences are smaller.