Well stool being stool and stool is even more odd! (Look them up if you don't know both of the meanings)
Bank and bank aren't that weird in both cases something is put "on" it
English has thousands of words with more than one meaning. Sometimes clearly related sometimes only when you look a little closer and dig a little deeper and sometimes not related at all, just a coincidence that two words ended up looking the same.
The word embankment is related. A raised area. Same goes for sandbank.
And ofcourse English still has bench that meaning precedes that of financial institution which is pretty recent (some here after the middleages). It originated from the word bench in the meaning of counter (not counting but the flat desklike surface)) first only for the counter itself later as a pars pro toto. Like wheels can mean a car
What they all have in common is the idea of a shelf. Something raised on top of which something can sit.
This is a germanic word that has meant bench (as in a raised but wide area) from the beginning it's very interesting that romance languages have it aswell. In this case the germanic words were there first. (The initial words with the initial meaning. As far back as proto germanic)
The use of the word bank/bench for the institution will have come from old italian
• originally germanic word meaning raised area
• turned into a flat raised surface area like in workbench
• Italian loaned that word from germanic and slowly it became to mean the entire institution (or building) nit just the counter
• other countries loaned this sense from italian. (In dutch directly in English possibly via French)