It depends on the medium - SO is closer to a wiki of questions, and Wikipedia has similar rules against pleasantries or signatures on article pages. (They do allow it on "talk" pages, and SO similarly has less restrictive rules in places other than their primary questions and answers.)

When you submit a documentation PR to an open-source project, do you put your name (real, screen or otherwise) in the section of documentation that you add? No; you let Git put it in the commit history. (And if you have to discuss it, you put it in the discussion, or let a forum system such as GitHub's put it there for you.)

Stack Overflow is the actual document being worked on, not the discussion thereof. (That's why there's a separate meta.stackoverflow.com domain.) And analogously to the Git commit history, your name is automatically stamped onto your posts, along with a link to your site profile, in a box at the bottom of the post.