Why does Safari ask for keychain password when it stores none?

Try setting up another admin user account to see if the same problem continues. Please post back on whether or not this worked. Also try the Safe Mode. Please post back on whether or not this worked.

Isolating an issue by using another user account 

Safe Mode - About

If it works in the Safe Mode, try running this program when booted normally and then copy and paste the output in a reply. The program was created by Etresoft, a frequent contributor.  Please use copy and paste as screen shots can be hard to read. Click “Share Report” button in the toolbar, select “Copy to Clipboard” and then paste into a reply. This will show what is running on your computer. No personal information is shown. If the log won’t post, try posting it in Pastebin and provide a link in a reply        Pastebin

Etrecheck – System Information

You may  have adware/malware installed. Try downloading this program which was written by Thomas R, a long time poster. The program will search for malware/adware. The program will do the work for you which makes it easy. It may be necessary to run more than one scan. After running the scans, restart the computer.

Malwarebytes Anti-Malware for Mac      10.10 and later

Malwarebytes uninstall

You are welcome.