One of my clients was getting a lot of these error messages in the log, but users were able to log in with no problem. It turned out that several of their databases had an auto-login set on the File --> File Options menu. That auto-login was for the Admin user, with no password, which is the default for new files. Since no user named Admin existed, that login attempt failed, causing the authentication error in the log. But the user-provided credentials were okay, so the user logged in just fine.
Hope that helps!
This clued me in . . not quite what you were describing but close.
We have an "opener file" for staff at the office, which simply takes users to the standard FMP login screen of the main database.
So, when a staff person logs in, they double-click to open their opener file, that file has the "Admin / BLANK" set up for the auto-login. The opener file's On-Open script opens the main database. FMP then tries to apply those credentials ( Admin / BLANK ) to the main database, and they fail. Hence the error. Meanwhile, the staff is presented with the standard login screen, logs in, and all is well.
So that answers that question . . .