How can I stop my iPhone from suggesting saved passwords?

Have you set 1Password as the default password manager on your phone?

If not, iOS presumes you're going to use Passwords.

To change it in iOS 18, go to your iPhone settings, General > Autofill & Passwords

From there you can turn off Passwords and only leave 1Password on.

In my example I use Chrome for work and Passwords for personal, so I leave both on and pick between them.

If you turn off Passwords it should stop asking you to save passwords to the Passwords app.

When you are asked “Do you want to use your saved password?”, does that appear in a box something like this? (The Passwords app logo near the top will certainly not appear in iOS 16.)

If so, what happens if you press the key-lock icon in the lower right?

And if this is not the style box presented, might the password be stored locally in the NYT Audio app itself?

Does it help to do Settings > Passwords > AutoFill Passwords = “off”?

UPDATE: It does appear tapping the key shown in your screenshot may be the solution!

Here is the pertinent information from the iOS 16 version of the iPhone User Guide. That’s what led me to believe tapping the key might work. They offer additional suggestions.

View your passwords and related information on iPhone - Apple Support