There's an argument that the structurally correct but semantically bizarre sentences are very helpful language-learning tools, and I'm inclined to agree with it. I have no objection to this sort of sentence - it's memorable and it teaches grammar divorced from meaning. I'm not trying to learn a phrase book after all.
I don't mind sentences with counterintuitive content. But I do wonder whether the ownership dispute of the tea is between my cat and you, or my cat and your cat. I'm inclined to think both are possible. In angrezi, it depends on where you put the accent... this is MY cat's tea, not yours vs. this is MY CAT's tea, not yours. I wonder if the same is true in Hindi.
I agree with you, I have seen this kind of ridiculous sentences in different languages from Duolingo. I cannot object the grammar but as a foreigner seriously trying to learn the language I question the utility of this. Even if I know the words or the structure my brain is saying "It is not possible".
I do not understand why "This is the tea of my cat, not yours" is incorrect. I learnt English as a foreign language and was taught that "'s" is to be used only with persons. So for all the Hindi learners that do not have English as a first language, I am of the opinion that this answer should be correct ;)
Grammatically, "This is the tea of my cat, not yours" is fine, although perhaps, as Sam362597 has said, somewhat stilted. However it implies that this tea belongs to my cat, not your cat, whereas "This is my cat's tea, not yours" implies that this is tea belongs to my cat, not to you. In spoken language, stress could deal with this, and the more usual pattern would be to talk about "my cat's tea", stressing "my cat". We (native speaker) use the possessive "'s" for all animate, and often inanimate objects quite freely.