MyEtherWallet gives you the ability to generate new wallets so you can store your Ether yourself, not on an exchange. This process happens entirely on your computer, not their servers. Therefore, when you generate a new wallet, you are responsible for safely backing it up.
MyEtherWallet.com & MyEtherWallet CX are not “web wallets”. You do not create an account or give them your Ether to hold onto. All data is created on our computer/your browser, not their servers. It’s not a website where you put your Ether and they store it for you. They just make it easy for you to save your wallet information in your browser/on your computer. Then they give you a place where you can access that information and do stuff with it.