Seems like you might be missing a setting for the user(s) or the global settings. Any settings you change are global, not site specific. I'd suggest making sure you have your settings right.
Settings>Security & Membership>Authentication and check Enable multi-factor authentication. Leave the "required" field unchecked otherwise it will require it for all users. Then go to a user you want to require MFA on and check the "Multi-factor authentication is required" box and save the user. Log out and try your MFA again.
There are no other settings you need to enable to require the webpart to use MFA, checking the box in Settings will do what you need. Good luck!