In ArcGIS Desktop/ArcMap, "Database Connections" is an application alias for the ArcCatalog folder in a users roaming application data directory. I say it is an application alias because only the application and its tools are aware of it. Although ArcGIS Pro still has roaming application data directories for users, database connection files are not stored their unless they have been marked as "Favorites," and even then it is a copy at a point in time of the original file.
Although "Database Connections" worked in ArcMap, I would argue the best practice is to give a fully-qualified path to the database connection file and not rely on a user-specific, application-level alias. If you are using fully-qualified paths, the tools will work the same between ArcMap and ArcGIS Pro.
If you want to access a database connection file that is part of a project, the database connection files are stored in the project's "home folder" which can be accessed via the homeFolder property of an ArcPy Mapping project:
import arcpy
import os
aprx_file =
GDB_directory = r"C:\Python Testing"
aprx = arcpy.mp.ArcGISProject(aprx_file)
targetEntGDB = os.path.join(aprx.homeFolder, "espr1sql14.sde")
extractedGDB ="600150be54a84779a461d4fc37247f91.gdb"
extractedGDB = os.path.join(GDB_directory, extractedGDB)
arcpy.Copy_management(extractedGDB, targetEntGDB)
Note: You really want to use os.path.join to build paths.