Fun fact: a few years ago there was a giant couch in the Eftelling sponsored by ABN-AMRO, which they called the "bank bank" :)
Yes, it can mean both just like there are loads of words that can have multiple definitions depending on context.
Bank is cognate with bench actually. Because the "bankers" back in the day would sit on benches while conducting business and thereby got their name. In Italian, banca for bank and panca for bench (Italy being where banking got started).
actually ''bench'' in french is ''banc'' (silent c) and ''bank'' is ''banque''