In ArcCatalog, or in the Catalog window in ArcMap, you add connections to folders, databases, and servers. In ArcGIS Pro, you can add connections in the Catalog pane, in catalog views, or from the Insert tab on the ribbon.

Connections in ArcGIS Pro are specific to the project to which they are added. You can use project favourites to add frequently used connections to new projects.
Common data management tasks such as copying, moving, and deleting items are performed in the Catalog pane or a catalog view. The Catalog pane is comparable to the Catalog window in ArcMap; a catalog view is similar to ArcCatalog, although it's not a stand-alone application.
Learn more about migrating data connections to ArcGIS Pro
Learn more about catalog views and the Catalog pane