If your install data script successfully installed your custom attribute, now you just need to override addition information .phtml file and set your custom attribute in that.
Create Vendor/TestModule/view/frontend/layout/customer_account_create.xml
<?xml version="1.0"?>
<page xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" layout="1column" xsi:noNamespaceSchemaLocation="urn:magento:framework:View/Layout/etc/page_configuration.xsd">
<body>
<referenceContainer name="form.additional.info">
<block class="Magento\Framework\View\Element\Template" name="additional_zipcode" template="Kage_RegisterForm::form\register.phtml"/>
</referenceContainer>
</body>
</page>
Create Vendor/TestModule/view/frontend/templates/form/register.phtml
<fieldset class="fieldset create account" data-hasrequired="<?php /* @escapeNotVerified */ echo __('* Required Fields') ?>">
<legend class="legend"><span><?php /* @escapeNotVerified */ echo __('Additional Information') ?></span></legend>
<p>
<div class="field regulation">
<label for="regulation" class="label"><span><?php /* @escapeNotVerified */
echo __('Additional Zipcode') ?></span></label>
<div class="control">
<input type="text" name="additional_zipcode" id="additional_zipcode" title="<?php /* @escapeNotVerified */ echo __('Additional Zipcode') ?>" class="input-text" data-validate="{required:false}">
</div>
</div>
</p>
</fieldset>
THANKS.