Possessive is usually denoted by apostrophe + s ("possessor's"), and its is an exception. Anyone who was never taught (or has forgotten) that it's an exception, will spell it it's instead of its. Though I haven't asked each and every person who misspells it if that is the case, I have come across this being the case, and I think it's a logical mistake.
As for your specific questions:
- Am I wrong in thinking that the only correct spelling for the possessive is its? Is it's an accepted spelling somewhere in the Anglosphere?
Proving that there is no one who spells a word differently (purposely) is not practical, so I assume this question is mainly part of your main one, you are offering a logical explanation as to why people would spell it differently. But since I think my answer above is logical, and since I haven't found any current dictionary (granted, my search was pretty cursory) saying otherwise, my answer would be: No. You are not wrong.
- Are people not aware of this, or they do not care?
Again, I didn't ask everyone, but those who I've corrected, were unaware. I doubt people who ensure their spelling of other words is correct, would have a case against one word.
- Is it simply a very widespread typo?
I think that the answer to No. 1 answers this too - this is unlikely (as you pointed out), and you are probably asking this only for lack of a better answer.
- Should I correct people in forums / comments about this?
Since this is a general question not specific to "its", and deserving a separate question, I am answering it in line with No. 1 again - Can you correct them or might it not be a correction at all? And as there - I think it is a correction, though, I don't know if correcting people's spelling mistakes is a good idea.
Side note: as mentioned in another answer here - it's actually was a correct spelling in the past.
its
? Because it's very easy to forget, and the contracted formit's
is used much more frequently, so it becomes a habit to add an apostrophe. This is more likely to happen with native speakers then with careful learners (in my experience).