No, has nothing to do with a Ponzi scheme. And yes, they relay on new customers.

Think about having a trading platform, and one customer gives you 10 BTC. He now can offer this BTC for any price, 500 Euro/BTC for example.

This will not matter as long as you do not find another customer, which gives you Euros. Without Euros no trade.

Two customers have an agreement about the price of one bitcoin, and if they agree they swap the currencies. This has no influence on the assets of the exchange platform, they earn the fees only. And they have to keep the deposits of the customers save and separated from their own assets.