what others think
Well, based on the situation as you described it in your comment here, this might have been a case of a bit more than just writing your own, better answer, and instead a case of not so welcome behaviour.
But the expected way to behave is to write your own, (more) correct answer when you see another answer is wrong, and linking back to or quoting that wrong answer and explaining exactly what was wrong with it, is not to be discouraged.
So I think this may be either a case of making a mountain of a molehill or a serious problem, sadly I can't see without a look at the answer that sparked this meta.
whether this sort of behaviour is widespread
I would answer this with 'no'. I've been moderating a site where there's been A LOT of bickering in comments over whether answers are right or wrong. I remember one of my own answers on that site, it has 90! deleted comments, all people trying to bicker over parts of the answer they felt were 'wrong' or 'right'.
Yet the question has 0 answers that link back to or explicitly mention the answer I wrote there, and while the bickering and sniping moved to meta, it did not result in any proper answers on main.
And that goes for most of the controversies I see on sites: The sniping is always happening in comments, and when those are removed, it seems to move to meta with questions like: Why were my comments removed? Why can't I say I disagree with an answer in a comment? Why is this answer still up?
I've only rarely seen people writing their own answers instead, explaining the parts that may go wrong when following the other answer and backing those claims up with facts, references, or in case of the site I moderate, relevant and analogous experiences. Usually, the bickering is a huge sign of people having opinions, but no answers.
whether critique of an answer should be limited to the comment section for that answer
So, my answer here is going to be 'no, but'. In the case you seem to describe here, whatever was in that answer might not have been fit for the comment section either, and that kind of critique should neither be in the comment section nor the answer.
But comments are there to request clarification, and to add information. A comment can be used to point out that an answer may have overlooked an essential part of the question, or request further clarification. Comments can be used to add or correct facts like which year something happened in, but they should never be used to say 'I agree' or 'I disagree'.
So some form of critique belongs in comments, some belong in other answers and another form of critique has no place on these sites at all.
whether cross-answer sniping should be discouraged explicitly in the guidelines
There's no need for special guidelines against writing answers that point out the flaws in other answers in a reasonable way. People are encouraged to write their own answers whenever they feel the existing ones are wrong and to explain why their answer is more right, they need to explain why another way of doing it is wrong.
But what you're describing in your comments seems like it may have been a bit more than 'just the facts'. We still don't need any new guidelines, as we already have guidelines to deal with unfriendly behaviour, and those include 'subtle put-downs and unfriendly language'.
If people go out of their way to trample another answer into the ground, lookt to these guidelines. You can edit the answer to remove the rant against the other answer, while making sure to preserve it's intent and leave the facts. If that doesn't work or seem feasible, you can ask other people to take a look, raise an issue on a site-specific meta. If there's a lot of unfriendly language going on you can still try to edit or flag the answer as rude if it's unsalvageable.