Magento 2: How to set custom template for core block

Just reference necessary block and set template as attribute of referenceBlock node:

<?xml version="1.0"?>
<page xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:View/Layout/etc/page_configuration.xsd">
    <body>
        <referenceBlock name="customer_form_register" template="VendorName_ModuleName::form/register.phtml"/>
    </body>
</page>

Alternative syntax:

<referenceBlock name="customer_form_register">
    <action method="setTemplate">
        <argument name="template" xsi:type="string">VendorName_ModuleName::form/register.phtml</argument>
    </action>
</referenceBlock>

Also if you are using Enterprise Edition, make sure to put Magento_CustomerCustomAttributes to sequence in your module.xml because it also overrides this template and may override your changes.