Um, I must admit this sounds extremely strange, but perhaps things were that different 20 years ago. Today, banks shouldn't even admit to someone having an account with them until they have IDed themselves. In my case I have my standard current account with UBS (UBS is one of the biggest retail banks in Switzerland, unlike abroad where there are no retail branches to my knowledge), and I can tell you from experience that until they have had a long hard look at my passport, they won't even talk to me (much less give me money from a friend's account).
As for my employer, which is a classic traditional Swiss private bank, you would have your own relationship manager who knows you and follows you from you first enter the bank and for the duration of your relationship with us. All your contact with the bank, in person or otherwise, would go through him or her. I know of a case where a friend of mine's father was a client with another private bank, and she'd been meeting the relationship manager several times with her father in the past. Nevertheless, one time she was in Switzerland she decided to visit the bank to check on the account's performance, but as she was not formally listed as a co-owner of the account, the relationship manager refused any information, despite recognizing her.
I think your friend's experience was rather unusual - as a matter of fact, if your friend came to me tomorrow, and I volunteered information on your account, I would be guilty of revealing client information which carries a prison sentence as well as a nasty fine.
The Swiss are extreme when it comes to keeping people's privacy. Even your tax payments (tax at source/quellensteuer) is anynomized before your company hands it over to the tax authorities on your behalf if you have a direct taxation/tax at source arrangement.
A quick comment on the numbered accounts though: one benefit of the numbered accounts which I failed to mention is that even internally in the bank, only a handful of people know who hides behind the number, which means that even though you are on file with the bank and still must idntify yourself upon opening the account, the number of people who knows who you are is very very low, and not potentially all 20'000 employees of a large bank like UBS or Credit Suisse with access to normal client information. This is the solution Elvis would choose! ;)