As I indicated before, this problem is for Safari only. Chrome and individual apps (e.g. Amazon, Netflix and others) are working fine in remembering usernames and passwords. Until the problem is fixed, I recommend you to use the Chrome for the time being if you don't want to keep entering usernames and passwords every time.
Apparently, Apple didn't fix my original problem. However, somehow a pop-up message box appeared on September 24 that magically solved my problem.
Try the following:
Safari> press the "Clear History" and press "Clear Cookies and Data"
Safari>Passwords & AutoFills>Names and Passwords (Do this: turn it on)
Safari>Passwords & AutoFills>Saved Passwords (Check this: this should be empty)
Power down and turn off the iPad.
Power up the iPad
1. Open the Safari, go to one of your sites, enter username and password, press the "LogIn" button. (The pop-up
message box should appear immediately.)
If you don't get the pop-up message box, close the Safari and try Step 1 again. Again. Again. (For a few times.)
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That's what I did a few days ago when the pop-up message box appeared suddenly.
When Safari started remembering usernames and passwords, I went back to check the setting in Safari:
Safari>Passwords & AutoFills>Saved Passwords (Check this: this time it was populated with site addresses and usernames)