I do not get the popup to update my password when I change my password on a website using their update password form. The form usually has three fields, old password, new password and confirm new password. When the form is submitted, Firefox suggests to update the password but Bitwarden does not.
I know that it can be tricky if there are multiple logins for the same website. I would suggest using a dropdown/combobox that allows the user to select for which username to update the password when such a form is submitted. If there’s only one login, then there’s no need for the dropdown to appear.