It depends on the position of "sitt". "Dyret drikker sitt vann." would be correct, but this is not common to write, so you should always be writing it like in the above sentence.
The form runs as though we were saying "the water of hers" instead of "her water". Unlike English, the Norwegian preference is for the noun first.
I literally spent my entire schoolhood being told that I need to use it's when it's used possessively. Now I'm told that every school and exam board I ever went to was wrong?
I honestly never heard any teacher that "it's" would be correct in this situation. I guess the teachers you met referred to the general rule or they themselves don't know the rule.
I just wrote "it's" and it did not accept it. I use that all the time, and "nearly" all the time it's accepting it. So why not now?!
it's = it is = contraction of pronoun and verb
its = possessive pronoun
It depends per dialect, rather than per word. The Oslo dialect that the course teaches skips the T, but as long as your consistent, it's ok to pronounce the T as well.
Shouldn't the correct answer be "the animal is drinking it's water"? It with an apostrophe before the S is indicative of it being the possessive form, not "its".
Actually, you have it entirely backwards. "Its" indicates possession, whereas "it's" is a contraction of "it is". It/its works the same as you/your or him/his or them/their.
Not having apostrophes on possessive pronouns is probably my least favorite rule of English, but the others are: "yours", "hers", "ours", "theirs". Also, you wrote "It/it's" instead of "It/its" ;)