Politics, like mathematics, linguistics, economics, and other nouns ending in -ics, is syntactically singular. So the answer "a different politics IS possible" should be marked as correct.
I consulted it with a native speaker and she said that while 'politics' can take either singular or plural depending on the context, singular in this sentence sounds really weird...
In real world practice, I can't imagine anybody ever saying "Different politics are possible." anyway. "A different political solution/result/conclusion is possible" is somewhat more likely. "Different politics is possible." is even less likely even if syntactically more correct :-) .
"A different kind of politics is possible." or "Another kind of politics is possible."
I have the feeling that the proposed translation has not the same meaning as the polish sentence. I understand "Another policy is possible [than the one followed now]". The translation just says that there are different possible policies.
"Another policy is possible" just worked for me, although the meaning is a bit different.
A singular noun is certainly necessary after "another".
Perhaps "Other policies/politics...." might work, but I didn't try it.
"a different politics is possible" is perfectly normal. It might be an Irish thing...
A different politics is possible. was mine and it was rejected with - Different politics are possible., A different policy is possible. - being the two alternative answers given. But treating politics as a singular is very common in English. For example, this article from the British Academy, "Building a new politics?" In the same way as a new politics, a different politics would be used. If the Polish is meant to be only that a different policy is possible then that would be the different, but it doesn't seem to be from the translation here.