Echo show_count of categories separately without using wp_list_categories

Just call get_categories(). You’ll get back an array of term objects:

array(
    [0] => WP_Term Object
    (
        [term_id] =>
        [name] =>
        [slug] =>
        [term_group] =>
        [term_taxonomy_id] =>
        [taxonomy] =>
        Echo show_count of categories separately without using wp_list_categories =>
        [parent] =>
        [count] =>
        [filter] =>
    )
)

You can process this with wp_list_pluck to turn it into an associative array, so for example:

 $cat_counts = wp_list_pluck( get_categories(), 'count', 'name' );

This will return an array like:

array(
    'Geography' => 5,
    'Maths' => 7,
    'English' => 3,
)

For other taxonomies use get_terms() instead. get_categories() is really not much more than a wrapper for get_terms().

To show these like the picture you’ve added to your question, just loop over the array.

echo '<dl>';
// use whichever HTML structure you feel is appropriate

foreach ( $cat_counts as $name => $count ) {
    echo '<dt>' . $name . '</dt>';
    echo '<dd>' . sprintf( "%02d", $count ) . '</dd>';
    // use sprintf to specify 2 decimal characters with leading zero if necessary
}

echo '</dl>';