Within the Portal for ArcGIS Web Map you can add a web layer and then drilldown to the specific layer within that service and customize the popup. However, this functionality does not exist within ArcGIS Pro. When you bring in a web layer into ArcGIS Pro and then drill down to the specific layer to right-click, you are not presented with a context menu.
Am I missing something here? Why can I do this within Portal but not Pro?
Are you looking for this? If so, right click on the layer to get the menu.
I am trying to do that against a web layer's layer. You sent a screenshot on a specific layer. Drag and drop over a web layer (map service) from your Portal instance and then right and right-click on an individual layer. See my attachment for the difference.