Add the tooltip on entity view too

Proof of Concept: add a formatter like

namespace Drupal\field_description_tooltip\Plugin\Field\FieldFormatter;

use Drupal\Core\Field\FieldItemListInterface;
use Drupal\Core\Field\FormatterBase;
use Drupal\Core\Form\FormStateInterface;

/**
 * Plugin implementation of the 'description_tooltip_label' formatter.
 *
 * @FieldFormatter(
 *   id = "description_tooltip_label",
 *   label = @Translation("With Description Tooltip on Label"),
 *   field_types = {
 *     "string", "text", "integer", "entity_reference", "boolean", "list_string"
 *     // Add more types or use BaseFieldDefinition if general.
 *   }
 * )
 */
class DescriptionTooltipLabelFormatter extends FormatterBase {

  /**
   * {@inheritdoc}
   */
  public function viewElements(FieldItemListInterface $items, $langcode) {
  $elements = [];

  $description = $this->getFieldDefinition()->getDescription();
  $label = $this->getFieldDefinition()->getLabel();

  // Render all field values together.
  $values = [];
  foreach ($items as $item) {
    $values[] = $item->view($langcode);
  }

  $elements[] = [
    '#type' => 'inline_template',
    '#template' => '
      <div class="field-tooltip">
        <div class="field-label" title="{{ description }}"><strong>{{ label }}</strong>:</div>
        <div class="field-items">
          {% for value in values %}
            <div class="field-item">{{ value }}</div>
          {% endfor %}
        </div>
      </div>',
    '#context' => [
      'description' => $description,
      'label' => $label,
      'values' => $values,
    ],
  ];

  return $elements;
}