For starters, the correct code is this:
add_filter( 'generate_navigation_search_output', function( $form ) {
$form = '<div class="navigation-search">';
$form .= do_shortcode( '[searchform dropdown="category"]');
$form .= '</div>';
return $form;
} );
This way, the function will override the original search form instead of creating a second search form.
If you want to modify the search form, you can do this instead:
add_filter( 'generate_navigation_search_output', function( $form ) {
$form = '<div class="navigation-search">';
$form .= <<<END
<form action="POST"...
the rest of the form HTML code here
</form>
END;
$form .= '</div>';
return $form;
} );
You can take the Relevanssi search form HTML code from the page and paste it in the function. Then you can easily change the parts you want to change: remove categories, change texts and so on.