Translation:The city's bank is very rich and famous.
Is there a difference between a városnak a bankja and a város a bankja? How come dative is not used sometimes?
If you mean "a városnak a bankja" and "a város bankja" (without the second 'a') then my answer is nothing. :)
so it is wrong to have a definite article for the object being possessed?
Without using "-nak", it is. "a város a bankja" actually means "the city is his/her/its bank"
This is a weird sentence in English. People are rich and famous, whereas banks are well known and successful, I would suggest.
Yeah, I guess well known instead of famous makes sense, but a "rich" bank might imply some statement about worth on the stock exchange market? Or high equity ratio? Or it is "too big to fail"?