That looks like it's easy to work around:
1. Deposit from one address (your wallet) and verify ownership by sign message or something - do they even allow that?
2. Now you can withdraw money from any exchange, mixer, website, or dark money service, send all the withdrawals to another address, transfer the entire balance of said address or part of it to the verified address and then deposit.
3. Bitstamp won't say anything (but it remains to be seen if they are that gullible. My sources say no.)
This means you cannot use an exchange wallet as your main wallet, receiving and sending funds directly from it. When you deposit and withdraw from an exchange you must first send to a specific wallet, or sign a message from that wallet.
It has many privacy and usability functions, it's a hassle, and it is done mainly to "protect" the exchanges. At the expense of users, of course.