Direct access to the currency exchange market

Is my observation that the currency exchange market is indirect correct? Is there a particular reason for this? Why isn't currency traded like stocks?

For stocks it is pretty simple where the stock is held with a depository. Hence listing matching is simple and the exchange of money is via local clearing.

Currency markets are more global and there is no one place where trades happen. There are multiple places where it happens and is loosely called Fx market place.

Building a matching engine is also complex and confusing. If we go with your example of currency pair, matches would be difficult.

Say;
If we were to say all transactions happen in USD say, and list every currency as item to be purchased or sold. I could put a trade Sell Trade for Quantity 100 Stock Code EUR at Price 1.13 [Price in USD]. So there has to be a buy at a price and we can match. Similarly we would have Stock Code for GBP, AUD, JPY, etc. Since not every thing would be USD based, say I need to convert GBP to EUR, I would have to have a different set of Base currency say GBP. So here the quantity would All currencies except GBP which would be price.

Even then we have issues, someone using USD as base currency has quoted for Stock GBP. While someone else using GBP has quoted for Stock USD.

Plus moving money internationally is expensive and doing this for small trades removes the advantages. The kind of guarantees required are difficult to achieve without established correspondent bank relationships.