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Thank you for the detailed explanation.
Based on the Microsoft documentation(https://learn.microsoft.com/en-us/azure/artifact-signing/), custom asset types in Microsoft Purview are created through the Apache Atlas type system. Microsoft documents how to create custom type definitions, define relationships between custom types, and initialize assets of those custom types in Microsoft Purview.
Microsoft also documents that assets created through the REST API can be searched and browsed in Microsoft Purview Unified Catalog.
However, I could not find documentation that describes a supported way to register a custom asset type as a taxonomy-supported source type for the same hierarchical UI filtering experience used by Azure Databricks Unity Catalog.
For Azure Databricks Unity Catalog, Microsoft documents built-in support for extracting technical metadata such as metastore, catalogs, schemas, tables, columns, and views. That appears to be part of the supported Databricks Unity Catalog connector experience, not a documented extensibility point for custom Atlas types.
Because of that, even if the custom Starburst entities, relationships, and indexed attributes are created correctly and are searchable through the generic catalog search APIs, I would not assume that the Purview UI will automatically render the same Instance / Catalog / Schema filter experience that is available for Databricks Unity Catalog.
As a practical workaround, you may consider using Purview collections to organize assets into a hierarchy that matches your Starburst structure, because Microsoft documents collections as a way to organize assets and sources into a hierarchy. This would not be the same as the Databricks Unity Catalog hierarchical filter UI, but it may help with browsing and governance organization.
If you need the exact same hierarchical filtering behavior in the Purview UI, I would recommend opening a Microsoft support request or submitting product feedback, because I could not find a documented API or configuration option that enables this behavior for custom asset types.
Hope this helps.