No, there shouldn't.
Documentary is less of a medium than rather a genre or style. Documentaries can very well be either movies or TV-shows, which are the media that we separate ID questions into. So in this case the working procedure is to tag the question with an appropriate ID tag (identify-this-movie or identify-this-tv-show), depending what it is. And then add the documentary tag on top of it, like we do with e.g. animation or horror (we don't have an identify-this-animation tag).
I understand that this approach has its disadvantages, too. For example the distinction between movies and TV-shows might not even be necessary and it also requires additional tags as helpers that otherwise wouldn't make too much sense of their own. But it's the currently existing tagging procedure for ID questions and as long as that doesn't get a major overhaul, introducing an identify-this-documentary tag will be a huge inconsistency in this approach.