Sure it is. Let’s leave out the fact that we see plenty of people using gold dragons – Dunk, for example. Let’s just do a thought experiment. 

A load of bread costs around three coppers. Assume that everyone in King’s Landing eats a loaf a day – which is pretty close to the statistics we have of ancien regime France, for example – that’s 1.5 million coppers that bakers are getting, which works out to about five thousand pounds of metal, on a daily basis. Trying to store that much metal is a giant pain, but it gets a lot easier when you can turn that 1.5 million coppers into 26,785 stags or 127 gold. I guess the point I’m getting at is that when you’re thinking about currency, in addition to thinking about different economic classes, you also need to think about different economic scales – even if most people aren’t throwing around gold on a regular basis, there’s a lot of economic activity in aggregate and having different denominations makes that easier. 

Now, as for why those denominations are metallic rather than paper…well, let me recommend Extra History’s series on the history of paper money:

The quick version of a very long and complicated history is that it depends on social and economic organization and levels of trust. With regards to the former, it comes down to this: how many people are there for whom this given note is going to be useful? Yes, merchants and nobles might be doing enough business that they’d be able to use a letter of credit from a given merchant, but what happens if that merchant or noble needs to buy a horse from a farmer who’s never been to King’s Landing, let along Braavos? 

With regards to the latter, before governments get into the business of printing money, you’re basically taking it on trust that the particular bank’s or merchant’s or nobleman’s notes are worth what they claim to be, and you run the risk that a bank failure or a bankruptcy or fraud (because the potential for fraud with letters of credit and other notes is huge) leaves you with nothing. 

And it gets even more complicated, because there’s no standardization, and so you have to discount the value of the notes of this merchant or this bank, which makes valuing the notes or making change or basic stuff like that really complicated. When I was in high school, my family sometimes went on trips to Old Sturbridge Village, a living recreation of an 1830s New England village. One of the places you could visit in the village was their bank, and I remember being shown these massive ledgers where bankers would record the different bank notes they have seen and what they think the notes were worth, and it’s a huge undertaking. 

Whereas gold coins have the advantage that they can be weighed and assayed to figure out how much the coin is worth, so you don’t have to do any of that. And as MMT theorists have pointed out, there was another advantage to coins: they had the government’s stamp on them saying what the coin was supposed to be worth, and they had a relatively steady value based on what the government would pay for goods and services, and how many coins they would accept as payment for taxes. 

That being said, the gold standard is insane, fiat money is a huge advance in human civilization, but it took a lot of work to make it possible.