"The professor's newspaper is in the professor's office."
Translation:صَحيفة اَلْأُسْتاذ في مَكْتَب اَلْأُسْتاذ.
I think you're right. I suppose you could argue that if it was the same professor you'd simply say "her office" or "his office" the second time, so there must be two professors! But that cuts no ice with me, it's an unidiomatic sentence in English and consequently impossible to translate idiomatically into Arabic.
I've given this a bit more thought, and looked closely at my screenshot of the two options, and I've come to the conclusion that while "the (male) professor's newspaper is in the (female) professor's office" isn't grammatically wrong, it wasn't what the question-setters were aiming for, so (I guess) fair enough that it is marked wrong. (By the same token I occasionally mistranslate "kariim" as the name Karim, which could be perfectly correct, but isn't the programme's intent, so is [quite rightly] marked wrong.) They can't write the checking algorithm to cater for all conceivable possibilities.
Because the possessive (expressed with the definite article in English) in Arabic is expressed by putting the possessed item in the indefinite (i.e. without article) followed by the possessor in the definite (i.e. with definite article ال unless it is a person or a place name).
It's defined by being the professor's, so it doesn't need to be defined by the definite article. The situation is s actually similar in English, although the word order makes it a little ambiguous — in "the professor's newspaper" there's only one article to cover both the professor and their newspaper.
EDIT — I meant "office", not "newspaper", but it comes to the same thing.