Thank you for your post.
Generally, when the password doesn't store in the keychain, then the keychain is corrupt. However, if the file can be opened in FileMaker Pro but not FileMaker Pro Advanced, something else is affecting the keychain. If you have any plug-ins installed, temporarily disable them in FileMaker Pro Advanced and try again.
In case the keychain is corrupt, go to your user Library, open the Keychains folder, and move the folder that contains a 36 alphanumeric name to the trash, and then immediately restart your Mac. The Mac will then create a new folder in the Keychains folder to replace the one that was just deleted. It will have a different 36 alphanumeric character name, and it will have new keychain databases inside.
Keep me updated with any progress.
Have you restarted the machine since installing the latest update? I had the same problem and restarting the client machine solved the problem.
The same thing has happened to me. Yesterday, databases I was working with were opening automatically via Keychain. This morning I updated to FM Advanced 13.0v4 and none of the files are opening automatically (ie I have to provide the account / pword details). It would seem that it is unlikely that this is a corrupted Keychain issue as a restart resolved the issue (as per Rick's suggestion).
Well, so much for my advanced troubleshooting skills! Rick, you nailed it on the head. A simple restart appears to have fixed the problem. And I agree with you, Jonathan. TSGal, thank you for your suggestions. But I have no plugins and I don't believe a corrupt keychain would be the issue when it still works for all other applications - including FMP.