Primitive Error when saving a custom field with a text_format widget
Here is what solved the issue:

I used messageFormValues in my Widget.php to override the Array with the text value, so it could be stored in my database.

public function massageFormValues(array $values, array $form, FormStateInterface $form_state) {
  foreach($values as $key =&gt; $value) {
    $values[$key][date_text] = $value[date_text][value];
  }
  return $values;
}


To unescape the html for my twig template(so it is displayed as html and not as text on the site) I used the raw attribute.

With this solution no Text-Format will be saved, but it was not necessary for my custom field anyway.