Is the adjective “vain” considered offensive when applied to a person?

The New Oxford American Dictionary’s definition makes it quite clear that it indicates disapproval (“excessively high opinion”):

vain (adjective)
having or showing an excessively high opinion of one's appearance, abilities, or worth: their flattery made him vain.

Vanity is one of the seven deadly sins. So when you accuse someone of being guilty of a cardinal sin you are rebuking them for a serious flaw in their character.

By the way, you are greedy and lustful.

Oh, I didn't mean you.

As far as I know, vanity is a negative trait in any cultural context. And thus, "vain" is an insult to anyone who understands the word.

It is often used as quite a mild insult, I think (like a lighter, relatively benign form of arrogance), but can also be used a quite harsh description.

I don't know about "dangerous", but I would advise against using the term unless you're trying to call someone vain (obviously)