Pro uses Python 3.x, ArcGIS Server still uses 2.x (2.7 I believe). The .mp module is new to Pro
And although you didn't specifically mention publishing to Server from Pro, that is not supported. Check this thread out
also tagging ArcGIS Enterprise for more exposure.
According to the documentation (The Python 3 runtime for ArcGIS Server on Linux—ArcGIS Server Administration (Linux) | ArcGIS Enterp... ) the runtime for 10.5 moved to python 3.5 ..
In my code I call the new arcpy.mp API, the _mapping part which is not found is called in the background
Pro using arcpy.mp http://pro.arcgis.com/en/pro-app/arcpy/mapping/introduction-to-arcpy-mp.htm and python 3.5, ArcMap uses arcpy.mapping and python 2.7, but the version of python shouldn't be an issue (I think).
arcpy.mp is accessed through Pro's anaconda distribution. No clue how or what server uses, let alone Linux