How to check specific value in two metabox?

You need to add the meta_compare argument to the meta query . Now your code will be like :

 'meta_query' => array(
                        'relation' => 'OR',
                        array(
                                'key' => 'pb_responsible_home_person',
                               'value' => $current_player_name,
                               'compare' => '='

                        ),
                        array(
                                'key' => 'pb_responsible_away_person',
                                'value' => $current_player_name,
                                 'compare' => '='

                        ),

            )// meta query end 

EDIT

$current_player_name = get_post_meta( get_the_ID(), ‘football_team_team_name’, true );

    $m_keys =array('pb_responsible_home_person','pb_responsible_away_person')

    $meta_query = array('relation' => 'OR');

    foreach ($m_keys as $m_key) {
        $meta_query[] = array(
            'key' => $m_key,
            'value' => $current_player_name,
            'compare' => '='
        );
    }

    $args = array(
       'post_type' => 'football_fixture',
       'meta_query' => $meta_query
    );

Try this, working when i checked locally