What right do exchanges have to know such details anyway? If a trader has the capacity to trade, then let them be. They already know the details of the person. Just chase after the said person should something shady occur with that person's account, no need to learn all these unnecessary stuff.
I doubt it's Bitstamp itself coming up with this, but more so a requirement of their banking partners. I also think that this will be replicated by other exchanges in the coming few years.
Banks provide a service to exchanges, which technically means that they too are participating in money laundering and whatnot would criminals use them as a way to deposit or withdraw whatever fiat currency.
We just have to accept that the centralized ecosystem will become more firmly centralized and controlled as time goes by. It shouldn't come as a surprise to people. Crypto is money and easy to move.