Search for two strings in WP search

I am not 100% sure what you are trying to accomplish, but you can “search for both strings” simply by concatenating them:

function checkForAa($query){

    if($query->is_search){

        $search_string = $query->get( 's' );

        if( stripos( $search_string , 'aa' ) !== false){
            $new_string = str_ireplace( 'aa' , 'xxx' , $search_string );
        }elseif( stripos( $search_string , 'å' ) !== false){
            $new_string = (str_ireplace( 'å', 'aa' , $search_string ));
        }

        if( !empty( $new_string ) ){
           // Concatenate the strings
           $query->set( 's', $new_string.' '.$search_string);
        } 

    }
}

add_filter('pre_get_posts', 'checkForAa');

$s="aabcd";
$query1 = new WP_Query( 
  array( 
    'posts_per_page' => 3, 
    's' => $s
  ) 
);
var_dump($query1->request);

This kind of search will become very inefficient, very quickly, with long strings though.