The flag of Amsterdam was there before the coat of arms. Don't get me wrong, we love both of them. But for interest sake, look up the history of the coat of arms - there is a lot to be proud of.
Agh, Amsterdam can be proud of many things. However, her coat of arms is probably based on another city (region) in the Netherlands. Those three crosses (Andreas crosses) have been used since 1203 in ... yes, the lord of Breda. A region that had much more influence than Amsterdam in the Middle Ages.