I've done some research into the matter and here's what I found:
1. Bitstamp is incorporated in the UK and, based on FCA/HMRC statements made in 2013, Bitcoin businesses in the UK do not require any specific license. They have no legal obligation to request KYC/AML information to their clients, as far as British law goes. They are requesting it because they want to, as confirmed on the Bitstamp site: "Whilst Bitstamp are currently unregulated and do not fall with the scope of the AML/CTF obligations in the UK the senior management have implemented systems and procedures that meet the UK AML legislation."
2. Bitstamp - much like every other exchange - uses a traditional bank, Unicredit in Slovenia, to conduct their operations. Unicredit has a legal obligation to subject Bitstamp to KYC/AML, and they're going to be fined in case Bitstamp does something dodgy (see the 1.9bn fine paid by HSBC for not monitoring their clients as they should have done).
3. This has nothing, repeat nothing, to do with the fact that Bitstamp is a Bitcoin-related company. There's no EU, euro area or Slovenian law governing Bitcoin businesses. The ECB and the EBA issued warnings about Bitcoin, but a warning is not regulation. Probably there will be regulation in the future, but right now there is not. Any requests made by Unicredit to Bitstamp are simply in compliance to obligations a bank has with respect to any customer, Bitcoin or not.
4. Bitstamp - much like every other company - would incur sanctions if they were discovered to be involved in money laundering (duh). Again, that's not because they're a Bitcoin business. If the authorities discover someone is laundering money through a real estate business, they'll go after them (duh #2).
So where does this leave Bitstamp users? Here's some options:
a) Bitstamp's bank is subjecting the company to extended KYC and Bitstamp is in turn asking customers for the necessary data.
b) Bitstamp is anticipating new regulation (e.g. they believe they will be required to apply for a money transmitter license in the UK or similar) and they're trying to gather what they believe to be necessary documentation to be fully compliant with UK law as it applies to money transmitters.
c) Bitstamp suspects some customers are laundering money through their services and they want to kick these people out before the subpoenas start coming, leading to account freezes, and hence damaging image and revenue. They have a right to do so; they're not obliged to provide their service to anyone.
These are the good options which should leave all users pretty much relaxed, provided they're not doing anything awful. Of course the questionnaire is pretty invasive and some Bitcoiners might conclude it's at odds with crypto ethics; in that case they should just move their business elsewhere, much like a vegan would not finance a meat-producing company.
However, there are not-so-good options too.
d) Someone at Bitstamp is building a new service / product based on the wealth of data they have, e.g. anything along the lines of data mining for marketing purposes, and they're gathering as much information as they can so that they can improve the service / product and sell it for a higher price. Not particularly scary, although this would violate the privacy policy in case a third party is involved in the handling of personal information ("Your information, whether public or private, will not be sold, exchanged, or given to any other company for any reason whatsoever, without your consent, other than for the express purpose of delivering the purchased product or service requested by the customer"). It would be borderline legitimate if they're using the service / product internally, as per "We may use your Personal Information for the following purposes:
To personalize your experience (your information helps us to better respond to your individual needs);
To improve our website (we continually strive to improve our website offerings based on the information and feedback we receive from you);"
e) Bitstamp is undergoing financial difficulties, maybe connected with the recent downward swing in prices (they overextended themselves? They fractional-reserve'd customer money without telling anyone and now they can't make ends meet anymore?), and they're hiding between KYC/AML to buy time.
Also, I wouldn't underestimate the abilities of British law enforcement. This is neither here nor there - personally, if I were a Bitstamp user I wouldn't be bothered by the police reviewing my data/transactions to see whether I have been giving money to terrorists or mobsters, but I understand that many members of this community would object to this as a matter of principle even if they don't have and never will have anything to do with criminal activity.