Extending ArcGIS Metadata Style?

One Approach is to use ArcGIS Metadata toolkit.

https://support.esri.com/en/download/7511

To do this You can

• Document the XML format in which an item’s metadata is made available to the metadata editor.
• Change the manner in which metadata is displayed.
• Change the settings used to populate the metadata geoprocessing tools when buttons are clicked in the Description tab.
• Specify the Translator used to export metadata to a standard XML format.
• Specify if and how the metadata may be validated using an XML Schema or XML DTD.
• Change the pages included in the metadata editor, except for the Item Description page, which must be included.
• Change the values provided in drop-down lists in the metadata editor.
• Change the validation rules associated with the content that can be provided in the metadata editor.
• Change the elements included in an existing metadata page, except for metadata elements that are required by ArcGIS software.
• Add a custom metadata element to a page and to the ArcGIS metadata format.
• Add a custom page to the metadata editor.

Second approach is to use Geoportal Server. It is an open source product which allows meta data customization and has integration options available with Esri ArcMap and Server and Portal. It allows you to create your own standard. There is option to re-use and extend existing standards.

Required steps as describe on their github

https://github.com/Esri/geoportal-server/wiki/Add-a-Custom-Profile#profile-customization-process-step-by-step

• Develop a plan for your profile
• Copy the required files
• Update your template.xml file
• Update your definition.xml file
• Update gpt.properties
• Define how elements are indexed for search
• Update schemas.xml
• Finishing your profile

For this product details refer to their Github Page

https://github.com/Esri/geoportal-server

Esri page for this product

https://www.esri.com/en-us/arcgis/products/geoportal-server/overview