How to return only one instance of each, from the entire loop

You’ll need a loop before you get to your sidebar loop to initiate your unique values. You could run the same loop twice, with rewind_posts() between the two loops ( so you get your original loop, starting from index 0 ).

In your first loop, loop through and add all your values to an array so that the array looks something like this:

Array(
    [0] => '7011 Trondheim',
    [1] => '7048 Trondheim',
    [2] => '7022 Trondheim',
    [3] => '7011 Trondheim',
    [4] => '7011 Trondheim',
    [5] => '7048 Trondheim',
    [6] => '7022 Trondheim'
)

Note that it will have duplicate values, which is fine for this case. Next let’s count our values using the PHP function array_count_values() which will give us the results:

Array
(
    [7011 Trondheim] => 3
    [7048 Trondheim] => 2
    [7022 Trondheim] => 2
)

Now we have a unique array which also has a count of duplicates!

NOTE

I’m not familiar with ACF but I imagine we could do something like this to achieve what we’re trying to do – also not tested:

<?php
    $addresses = array();
    if( have_posts() ) {
        while( have_posts() ){
            the_post();
            $city = get_field( 'poststed' );
            $zip = get_field( 'postnummer' );
            if( ! empty( $city ) && ! empty( $zip ) )
                $address[] = "{$zip} - {$city}";
        }
    }

    if( ! empty( $addresses ) ) {
        $uniqueAddresses = array_count_values( $addresses );
    }

    rewind_posts(); // Incase needed later.

if( ! empty( $uniqueAddresses ) ) : ?>

    <h3>Filter etter postnr:</h3>
    <ul>

<?php foreach( $uniqueAddresses as $address => $count ) : // Let's loop through our array 
        // Let's split our unique address back in to `poststed` and `postnummer`
        // $addressArr[0] => postnummer
        // $addressArr[1] => poststed
        $addressArr = explode( ' - ', $address );
?>

        <li><a href="https://wordpress.stackexchange.com/questions/168029/?postnr=<?php echo $addressArr[0]; ?>"><?php echo "{$address}({$count})"; ?></a></li>

<?php endforeach; ?>

    </ul>

<?php endif; ?>