How to set variable to specific field when querying

Always sanitize input fields. Use $wpdb->get_var to check a single variable. You forgot to add the compare value to the WHERE clause.

function username_auth() {
     global $wpdb;

     $username = sanitize_text_field( $_POST['custom_field_username'] );

     $user_login_sql = $wpdb->get_var("SELECT username FROM registration_info WHERE username="$username" ");

     if( $username !== $user_login_sql ) {
         wc_add_notice(('Incorrect username123'), 'error');
     } 
    
}