It is possible but will require custom code. You will need to show groups and give users functionality to change their groups.
If you know php i can guide you, or you can hire any developer, Or you can contact us using https://www.shamimsplugins.com/hire/

First remove group settings from user settings page using following code

add_filter( 'fep_form_fields', function( $fields ){
    unset( $fields['fep_groups'] );
    return $fields;
}, 99);

Then show a form where you want your users can set their groups. You can get all groups using Fep_Group_Message::init()->get_all_groups() and user selected groups using Fep_Group_Message::init()->get_user_groups(). When they submit their form determine if they added or removed any group and save value accordingly.
You can get all in action in class-fep-group-message.php file. See this file.

Hi again,
If you need help with custom code for your settings page you can contact me through https://www.shamimsplugins.com/hire/