Bankers do make an appearance in the book. The main bank in ASoIaF is the Iron Bank of Braavos All of the free cities have their own banks, but the Iron Bank is by far the most powerful. It is considered highly foolish to trifle with the Iron Bank, where it is often said, "The Iron Bank will have its due." If necessary, the Iron Bank will use its vast wealth to hire sell swords to get its money back, and has the power to support new Kings, thus changing the shape of the lands.
Spoilered appearances for each book.
A Game of Thrones
We first hear of the Iron Bank in the first book, Game of Thrones when Ned Stark learns from the small council that Westeros is in a lot of debt, much of it to the Iron Bank.
A Feast for Crows
Considering it is foolish to trifle with them, it may not surprise you that in Feast for Crows Cersei opts to delay payments to the Iron Bank (against the objection of Pycelle). An envoy, Noho Dimittis, is sent to King's Landing, but he is initially given the run around from Cersei, who sends him to speak with Lord Gyles a few times, before finally being sent away and told that the Iron Bank would get its payments when the rebellion had been quashed.
A Dance with Dragons
Another envvoy, Tycho Nestoris, arrives at the Wall to meet with Stannis. The Iron bank intends to support Stannis, on the promise that he will pay the debts that the Iron Throne owes the bank. The Iron Bank will have its due.
We also hear that
Ser Harys Swyft, the Master of Coin under Lord Regent Kevan Lannister, has written to the banks of both Pentos and Myr, to try and raise enough money to pay back the loans of the Iron bank. This clearly suggests that the Iron Bank is so powerful that it's worth taking another loan in order to placate them.
Clearly banking is a very important part of the Known World. However we only hear of banks from the Free Cities of Essos. We do hear of other sources of loans. We learn that the Iron Throne is in a large debt to Tywin Lannister. It does not make clear the terms of the loan, but it is unlikely that Tywin will lend money without the potential for gain, whether through interest paid or promise of power.
Update based on the revised question
Who produces the currency of the 7 kingdoms? It's definitely not the Master of Coin, considering what I saw.
I don't believe there's any mention of this, I'll continue looking.
Is the currency centralized at all in Seven Kingdoms, or does each kingdom have its own currency?
There is a single unified currency throughout the seven Kingdoms. This consists of three coins: the Gold Dragon, The Silver Stag, and the Copper Star (7-pointed, representing the Gods). This was established after the unification of the Seven Kingdoms, following the War of Conquest.
There is some slight exception to this, regarding the Iron Islands. The Iron Born (particularly the men) tend not to use money as much (“the Gold price”) and prefer only to trade the things which they took by force (“the Iron price”).
Why do the kings prefer to borrow money from some bankers (i.e. to borrow from each other, since the bankers don't have any money of their own from the start) instead of getting rid of the bankers once and for all and issue their own coins?
This is a very complicated question, and you could ask the same of our world. If you start to dismantle the financial institutions, then you start to create many enemies. Banks have their money and power stored in many ways, and once you try taking apart one piece, the others will find retribution. It is mentioned that the Iron Bank of Braavos will be willing to incur losses, and hire sell swords to get their money back. It's all about reputation. If you believe that a bank will come back with swift retribution, then you might prefer to pay them.
There's also an issue with revolt. Once you start dismantling your own laws, regulations and institutions, this leads to anarchy. For a real world comparison, consider the revolts and rebellions surrounding Charles II reign and his actions of dissolving parliament.