What is the name of the citation style that uses square brackets in superscript? Wikipedia uses this style - are there any other papers, journals or institutions recommending it?
It is the "editor's note."
Brackets indicate that their content has been added by someone other than the article's primary author. This is called an "editor's note" and is often labelled as such. Wikipedia is known for allowing a great number of people to add to or edit its articles (provided they have the proper citations) and for having a large and active number of in-house editors.
And yes, these are seen pretty frequently in print media, especially in magazines, usually when an editor wants to reply to a minor point within a long letter to the editor, or in an article, to refer the reader to another related article in a current or recent issue.