Looking for a moment at the ArcMap documentation on What are ArcGIS database servers?
Within ArcGIS software, the term database servers refers to instances of Microsoft SQL Server Express that have been enabled to store desktop or workgroup geodatabases.
You create geodatabases and perform other administrative tasks for database servers through the Database Servers node in the Catalog window or ArcCatalog. Performing the administration of the database server and its geodatabases through ArcGIS Desktop means there is no extra software or database administration expertise required for you to create and use these types of geodatabases.
The "database servers" connections are for creating and administering database servers, or really just SQL Server Express, that contain desktop or workgroup geodatabases. I think the larger question is, how does or even does ArcGIS Pro allow for creating and administering desktop or workgroup geodatabases? The answer is, not really.
A search on ArcMap documentation for "database servers" returns 62 results, and the same search for Pro returns 0 results. At a minimum, it indicates that desktop and workgroup geodatabases are not managed through "database servers" in ArcGIS Pro, at least not yet.
Searching on "desktop geodatabase", "desktop geodatabases", "workgroup geodatabase", and "workgroup geodatabases" yield similar results. A look at An overview of creating geodatabases in ArcGIS Pro shows two subsections, one for a file geodatabase and one for an enterprise geodatabase. It is clear the initial focus of ArcGIS Pro is on file and enterprise geodatabases.
So, where does that leave desktop and workgroup geodatabases in ArcGIS Pro? Great question, and one that I have no idea in terms of an answer.