I don’t think the discussion is primarily about a huge problem anyone is having at all, so much as it about talking about cognitive tools for understanding a phenomena that is encountered in greater and greater degree the more broadly you are known of beyond the circle of people who simply know you.
For some people it certainly does become a problem. For some the idea of authorial construct is a handy tool that allows them to separate from their work. For some it’s simply a fascinating cognitive phenomena. It’s also important to note that it’s not only or even mainly about a person’s deliberate public persona.
In the case of authors, at least, people form opinions about who a writer is sometimes based entirely on what they’ve read in the writer’s books, and without any clues other than that and name.
This is one reason why several of my readers have been quite startled to find out that I’m a burly bald man and not the bookish woman they built in their heads by working with my gender-ambiguous name and the stuff of mine that they’ve read.