You are not going to get a definitive answer here. Talk to the mieterverband or the local commune.
Indeed, yes, some rooms are let out on that basis, some as Longbyt described, some to poor students who really do live there and shower at the gym or swimming-pool or at University or simply wash in their room using a jug of water from the hand-basin, and some to persons who wish to use the room to do officework or study, who may only occasionally stay over. Some premises are even let out without access to a toilet.
The important thing is that such conditions are specified clearly in advance, and in your case the landlord did so up front. Therefore, I doubt that you have any case to show that this arrangement could be in any way unfair or that it might compromise any rights you may have, or could be illegal. No. It is as it is.
The market situation in Zurich really is difficult, especially for people whose budget is small. I get that. It's a pity that this has led you to take a decision which isn't really what you want.
The worst case scenario is that you will have to pay the full rent for the full period of the rental notice in terms of the contract, from whenever you put this notice in writing and send it to your landlord be registered mail, or hand it to him personally.
Alternatively, you could set about trying to find a substitute tenant who would be willing to take over the contract any time from now.
I would say it is o.k. to exclude the use of the kitchen and maybe shower or tub. But in my opinion at least access to a loo should be there.
What is the situation regarding this?
Your case isn't just about the 2 months notice (which by law has to be at least 3 months for non-furnished living rooms such as yours btw) but just as much about cancel dates. For Zürich city the default cancel dates are end of March and September only, so unless the contract explicitly provides more options the first possible cancel date for you is end of March 2018, later if there's a longer minimal rent duration.
I would expect loo plus running water to be mandatory for living rooms, haven't found anything to that end though.
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It is his own duty to check if he is legally (permit, help with rent or whatever) allowed to do what he is doing, the contract will not disappear with this.
By talking to the Mieterverband (and giving them some money), he will be advised on the legality of the contract. How would that in itself get him into trouble? If the advice is "this is an illegal tenancy agreement", then the contract will be void. If the advice is "you cannot be denied use of the facilities", then the OP problem is solved - he enforces his rights. If the advice is "suck it up princess, it's legal and you signed" then he's stuck until he gives notice.
Besides that I cannot imagine this being illegal, also since it's not that rare, people want some extra money and thus rent out a room but don't to be bothered by it and thus keep a bunch of the general spaces for themselves. OP can go to the toilet, he can cook on his room, and wash himself.