You keep Iron coins at the House of Black and White. Description of iron coin is "so old and worn out you can no longer recognise his facial features (of the 'head' side)". You buy with that coin a trip to the House of Black and White anywhere in the world (Arya is literarly brought right to the front of the house, first by the ship across the sea then by boat over the cannals). It's said and repeated that Braavos was founded by escaped slaves of old Valyria. It is implied that the first Faceless man had important role in that endevour. Iron Bank is city (central, national) bank of Braavos. It is said that when a king won't pay for his debts, a country gets a new king. Same with all the other debts.
Of course, there's Jaqen and other FM insisting on paying life debts and valuing life and death. It is possible that two institutions are connected, simply by both being there from the start, from founding of the city of Braavos. Also, money talks, assassins listen. Arya got in trouble for killing a man without order and without FM getting paid for it.
It's also weird to note how weird "traditional" formal greeting in Braavos is. They might say hi and goodbye normally, but when you say "Valar morghulis" to them, their answer is "valar doheris". Approach a man from Braavos, say to him "All men must die" their answer is a formal "All men must serve". But only if you show an iron coin. And there's also Brusco's answer to Arya (the guy whose fish she was selling when she said that she must go serve): We all do.
So, I think the story of Braavos is like this: An assassin freed the slaves of ancient Valyria and founded a city with them. He was the best assassin in the world. The city got rich and powerful. The Iron bank was founded to keep the riches and make them multiply. He coined his own coins, from the iron chains of the slaves and he had put the face without facial features on them, to show that all people are equal. No slaves, no slave owners, no kings, no dragon lords. Every slave who ran away probably had chains or some other thing marking them as a slave, so they had at least a small start in life in his city. And every single person born in Braavos is indebted to the founder of the city, our assassin, for freeing their parents and their grandparents and unto 14th generation and making them a place where to be free. And Braavos hates Valyria and it's slave owners and dragon riders. Slavery is specifically forbidden in Braavos and every person found smuggling slaves or owning them is detained and executed. And the story of city's founding is that "other story, the one you do not share with anyone".