hook_forms() not working

What node_type_get_types() returns is an associative array of node type objects, keyed by the type. The description of the value returned is more understandable in the _node_types_build() documentation.

The hook implementation done from a Drupal core module, for example comment_forms(), helps to understand the correct code.

function comment_forms() {
  $forms = array();
  foreach (node_type_get_types() as $type) {
    $forms["comment_node_{$type->type}_form"]['callback'] = 'comment_form';
  }
  return $forms;
}

In your case, the code should be similar to the following one.

function my_dossier_forms($form_id, $args) {
  $forms = array();
  foreach (node_type_get_types() as $type) {
    $forms["my_dossier_node_{$type->type}_form"]['callback'] = 'my_dossier_form';
  }

  return $forms;
}

This code isn't much different from the code shown in the question, and it would not cause, alone, the class 'my_dossier_form' not found error.

The line 844 is the following one.

$form = call_user_func_array(isset($callback) ? $callback : $form_id, $args);

$callback is initialized from the following code.

if (!isset($forms) || !isset($forms[$form_id])) {
  $forms = module_invoke_all('forms', $form_id, $args);
}
$form_definition = $forms[$form_id];
if (isset($form_definition['callback arguments'])) {
  $args = array_merge($form_definition['callback arguments'], $args);
}
if (isset($form_definition['callback'])) {
  $callback = $form_definition['callback'];
  $form_state['build_info']['base_form_id'] = isset($form_definition['base_form_id']) ? $form_definition['base_form_id'] : $callback;
}

Line 844 could not think my_dossier_form is a class, except in the case $callback contains an array like array('my_dossier_form', 'mymodule_form');, or it contains a string like 'my_dossier_form::methodName'.

The warnings about an array to string conversion means Drupal is getting an array when it expects a string, for example from the following line (line 1094).

    elseif (isset($form_state['build_info']['base_form_id']) && function_exists($form_state['build_info']['base_form_id'] . '_validate')) {

I would just avoid returning a form ID that starts with $type as that is probably creating conflicts with other modules implementing hook_forms(), which are expected to return a unique form ID each. See module_invoke_all() to understand what happens when two hook_forms() implementations return information for the same form ID.

  foreach (module_implements($hook) as $module) {
    $function = $module . '_' . $hook;
    if (function_exists($function)) {
      $result = call_user_func_array($function, $args);

      // In the case of hook_forms(), $result contains an array.
      if (isset($result) && is_array($result)) {
        $return = array_merge_recursive($return, $result);
      }

      elseif (isset($result)) {
        $return[] = $result;
      }
    }
  }

To be more exact, two modules returning values for the same form ID could at least cause the call_user_func_array() expects parameter 1 to be a valid callback, class '[callback]' not found when they both set the value for the same callback. See the output of the following code, to understand what exactly happens.

$return["type_node_form"]["callback"] = "my_dossier_form";
$result["type_node_form"]["callback"] = "mymodule_form";
print_r(array_merge_recursive($return, $result));

The callback value becomes an array of two strings, which call_user_func_array() interprets as an array containing a class name and a method name.

Array
(
    [type_node_form] => Array
        (
            [callback] => Array
                (
                    [0] => my_dossier_form
                    [1] => mymodule_form
                )

        )

)

In this case, the conflict is with the Node module, which uses the following code for its hook_forms() implementation.

function node_forms() {
  $forms = array();
  if ($types = node_type_get_types()) {
    foreach (array_keys($types) as $type) {
      $forms[$type . '_node_form']['callback'] = 'node_form';
    }
  }
  return $forms;

As side note, the hook_forms() purpose is providing the same form builder for a group of forms whose ID follow a schema, such as in the case of comment edit forms, whose IDs are comment_node_[node type]_form. When provided, the form builder callback (passed as callback value) should build the full form, not part of it.
The fact my_dossier_form() builds just a submission button, and my_dossier_forms() is using the same code used by node_forms() makes me think the code purpose is altering the node edit forms, which should be accomplished by implementing hook_form_BASE_FORM_ID_alter(). That's what the Book module does with book_form_node_form_alter(), which is invoked for the node edit form of every content type.

function book_form_node_form_alter(&$form, &$form_state, $form_id) {
  $node = $form['#node'];
  $access = user_access('administer book outlines');
  if (!$access) {
    if (user_access('add content to books') && (!empty($node->book['mlid']) && !empty($node->nid) || book_type_is_allowed($node->type))) {

      // Already in the book hierarchy, or this node type is allowed.
      $access = TRUE;
    }
  }
  if ($access) {
    _book_add_form_elements($form, $form_state, $node);

    // Since the "Book" dropdown can't trigger a form submission when
    // JavaScript is disabled, add a submit button to do that. book.css hides
    // this button when JavaScript is enabled.
    $form['book']['pick-book'] = array(
      '#type' => 'submit',
      '#value' => t('Change book (update list of parents)'),
      '#submit' => array(
        'book_pick_book_nojs_submit',
      ),
      '#weight' => 20,
    );
  }
}

If then the hook needs to use the content type name, that is available in $form['#node']->type.