The exchange wallet is not the same as your wallet. So in the exchange wallet you work only with your public key/address, because you don't really have access to your private key.
In the case of a centralized exchange, you have neither a personal wallet, private keys, addresses nor public keys. You will own nothing basically. Once you hand over your coins to an exchange, it has full control over them. What you see on your balance is not what you have just deposited (real coins) but an entry in their database with the amount they owe you. They receive your coins, add them to their reserves and give you a promise to pay you back once you request. Like any other promise, the promise to pay you may also be broken without an exchange facing any legal consequences. Here is a great article I found that explains things in detail:
Buy vs Own: Do You Really Own the Crypto You Just Bought?
