It looks like you embedded the forum by hand using embed code rather than using the Muut panel in the page editor to Enable Forum. If you add the code by hand, the Federated Identities (login through your site rather than the Muut account) won’t work on that page. This section of the plugin documentation talks about the panel on the page editor. If you would prefer to only use Muut accounts, you can simply disable Federated Identities on the plugin settings (although you probably should still do the embed using the panel). Once you remove your code and enable the forum on the page using the Muut panel, the formatting may be different by default (it uses a template packaged with the plugin), but if it isn’t as you like, simply try dropping this line of code at the bottom of your theme’s functions.php file and it should do the trick. Let me know if not.
add_filter( 'muut_use_packaged_forum_template', '__return_false' );
The avatars then will be tied to their avatar on your site (or Gravatar by default). Let me know once you’ve changed that and if it isn’t perfect, we may have to do a little final jiggering to get it just right for you. ??