I see where I can add database and server connections via INSERT > Connections in the toolbar
But this seems to be project specific and are no longer there when I open a new project. What gives....how do I set this across the board where if I open a new project they are there ready to use?
Here is the documentation of the Favorites in ArcGIS Pro 2.1.x: Project favorites—ArcGIS Pro | ArcGIS Desktop
I think that will give you what you are looking for.
Geodatabase