How should I call a form in a theme?
As per the comments from Jaypan, the first thing to do is extending SearchBlockForm so you can give it a custom ID. Put the code in mymodule/src/Form/MymoduleSearchBlockForm.php.
Add the following code in the custom class (in this case, FoobarblkBlock) to create a custom block. Put the code in mymodule/src/Plugin/Block/FoobarblkBlock.php

/**
 * @file
 */
namespace Drupal\mymodule\Plugin\Block;

use Drupal\Core\Block\BlockBase;

/**
 * Creates a 'Foobar' Block
 * @Block(
 * id = "block_foobarblk",
 * admin_label = @Translation("Foo Bar block"),
 * )
 */
class FoobarblkBlock extends BlockBase {

    /**
     * {@inheritdoc}
     */
    public function build() {
      $form = \Drupal::service('class_resolver')-&gt;getInstanceFromDefinition('Drupal\mymodule\Form\MymoduleSearchBlockForm');
$form-&gt;setFormIdentifier('some_custom_value');

return \Drupal::formBuilder()-&gt;getForm($form);
}


Place the block in the required region.
The resulting form will be:

&lt;form action="/search/node" method="get" id="search-block-form-some-custom-value" accept-charset="UTF-8"&gt;