Customize user registration form by adding/modifying fields in the form and handling submit function
If it is not crucial for you to have the additional fields in custom code, I would highly recommend you to make use of the fact that user is a fieldable entity and you can add additional fields easily to it via administration at "admin/config/people/accounts/fields" - this way you do not need to handle saving of the data and display of form elements.

When doing so, you can still make use of custom validation or submit handlers or fiddle with the form and fields easily using the same technique you are using - hook_form_alter and add your submit handler to do whatever logic you require.

Back to your approach — the reason why your submit function is not being called is that you are assigning a string to it instead of adding an element to the array. Try this:

$form['#submit'][] = 'module_form_submit'; (notice the empty brackets on the left side).

If you want to completely overwrite the submit functions array use this:

$form['#submit'] = array('module_form_submit'); - but this will cause that the user will not be created and data saved (unless you do that by hand in your custom submit handler), this also cancels whatever some contrib modules to with the registration form, so you might have unexpected issues.

Edit after comments:

If you do want to keep the default behaviour, but do your logic before everything, you can put your submit handler as first, as they are executed based on their order in the array like this:

array_unshift($form['#submit'], 'module_form_submit');