I use 1&1, and I ran into the same situation a couple months ago: I was terminating one of my contracts, and they asked for my password over the phone to verify. To be clear: I was not closing my account, I was only terminating a single contract.
The way I "resolved" the matter was quite simple: as I am not stubborn, I just gave them my password. The person sitting on the other end of the phone call already certainly has godlike access to my account anyway, I am not stupid enough to use the same password for multiple accounts, and barring insanely epic hacks I know they are a real representative as I called them at their phone number; so, there is really very little to lose handing over my password to the customer support person.
In the end, rather than getting morally outraged and posting an article asking a question to an online community in the hope of unblocking your ability to conduct what is fairly simple business, you should just change your password when you are done with the call and move on with your life. It will save yourself a bunch of time and frustration.
Then, afterwards, if you don't like the way 1&1 operates (maybe you believe that this is indicative of a more underlying set of security mistakes, or maybe you simply don't agree with the practice and don't want to support it), you might then consider moving your accounts to a different provider: there are tons of people you can use to host servers, domains, or whatever else you may be using 1&1 for. However, it shouldn't block your ability to make things happen right now.