Style wp_link_pages as a select drop down list

Here’s one way to achieve that is by constructing the <option> tags with the wp_link_pages_link filter:

add_filter( 'wp_link_pages_link', function( $link, $i )
{
    return sprintf( 
        '<option value="%d" %s>%d</option>', 
        $i, 
        selected( $i, get_query_var( 'page', 1 ), 0 ), 
        $i 
    );
}, 10, 2 );

and then the <form> and <select> tags with the wp_link_pages filter:

add_filter( 'wp_link_pages', function( $links )
{
    if( empty( $links ) )
        return $links;

    return sprintf( 
        '<form method="GET" action="">
            <select name="page">%s</select>
            <input type="submit" value="%s"/>
        </form>', 
        $links,
        esc_html__( 'Go to page', 'mydomain' )
    );
} );

Here’s how it looks like in the Twenty Sixteen theme:

Hope you can adjust it to your needs!