Thank you for your post.
I am unable to replicate the issue. This is what I have done:
1. Under Mac OS X 10.9.5, I launched FileMaker Pro 13.0v3 and created a file "File.fmp12".
2. I went into File Options, turned off "Log in using", and clicked OK.
3. To verify this worked, I closed and reopened File.fmp12, and I am prompted for Account Name and password.
4. I entered "Admin" for the Account Name, left the Password field blank, and checked the option "Remember this password in my keychain". I clicked OK, and the file opened.
5. I closed and reopened File.fmp12, and the file opens without being prompted for the Account Named and Password.
Let me know what I'm doing differently than you so I can replicate the issue.
It generally occurs just after you installed a FileMaker Pro update.
Restart your computer will almost surely solve it.