The quoted language (whch I have verified is the complete text in the relevant FAQ section on the linked web site) is not as clear as one might wish, or as a contractual provision should be.
It defines a period relative to a starting date, but does not define what that starting date is.
If the starting date were, say 15 March, then the month after next would be May, the end of the month after next would be 31 May, and the third working day after that would probably be somewhere between 3 June and 6 June, depending on what day of the week 1 June falls on, and whether there are any public holidays early in June . A "working day" is normally a day that is not a Saturday, a Sunday, nor a Holiday.
The quoted provision is odd, in that it gives the latest date when the tenancy can be terminated. Usually a notice provision for a tenancy gives the earliest date on which the tenancy can be terminated, with language like:
The tenancy can be terminated only 60 days or more after the tenant gives notice.
I notice that the above quoted provision seems to be a translation of the provision given in the German-language version of the FAQ, which reads:
Das Mietverhältnis kann ausnahmsweise vor Ablauf der vertraglichen Zeit nur unter Darlegung besonderer Umstände beendet werden. Die Kündigung ist spätestens am dritten Werktag eines Kalendermonats zum Ablauf des übernächsten Monats zulässig. Zur Fristwahrung ist der Eingang entscheidend.
I do not know German well enough to determine if there is some clearer meaning in this version.
It might be that the start date intended is the date when the student moves in, or is admitted, but that is not clear to me.
Update after the comment by Awais Mirza
It does seem that the English version was produced by Google translate or some similar computer translation from the German version, and I gather that in the process the meaning was significantly distorted. Indeed the translation should have been checked by a human fluent in both German and English who could compare the meaning with that of the original. Particularly as, according to the question, this same English text formed part of the contract that was signed.
If the school authorities based their understanding on the German text, it would seem that there was no meeting of minds, and if such a case came to court, it might well be held that the provision was not legally valid as a contract, because the two parties did not agree to the same terms.