"THIS IS THE WATCH OF ADAM" The answer should be accepted. This may not be the natural order of the phrase in English, but I think the answer is not wrong. We are testing the Polish, not the English. I think "THIS WATCH IS ADAM´S" is correct too.
Well, but the English has to be correct nevertheless. I consulted this with natives and got the results that it would be "so strange that it shouldn't be accepted".
On a separate note, "This watch is Adam's" is just a different sentence with a different subject, so that's definitely a 'no'.
The point of grammar is to describe how language is used, not to allow new structures that aren't used. This is just completely unnatural and that construction is used only for some very elite/formal things like "The House of Gryffindor" or something like that.
In fact, English makes an animate vs. inanimate distinction here, which should please Polish speakers.
Inanimate: The edge of the desk, the heart of the problem, the best of the bunch (vs. The desk's edge, the problem's heart, the bunch's best)
Animate: the cat's tail, Adam's watch, the goverment's watchdog (vs. The tail of the cat, the watch of Adam, the watcgdog of the goverment).
The forms in brackets are perfectly comprehensible but just not what native speakers normally use (except in poetry). They are often a giveaway in a very fluent non-native speaker.