Most (this is close, about half) stock trades are not on the listed exchange. If I sell stock and my broker has someone else wanting to buy that stock they are just going to internally match us up at the current price instead of going to the listed exchange. There are a lot of "dark pool" exchanges that brokers will check with first when someone wants to buy/sell stocks and so only rarely (again about half the time - not very rare) will they go all the way to an exchange.

Nothing stops an exchange from accepting trades for a stock that isn't listed there. (there may be local laws that say otherwise, but there are lots of different countries). However if you are not a listing exchange brokers might not think to check your exchange when someone wants to trade and so you won't get enough volume. If you are the exchange a broker checks first though you can be the exchange.