You can view maps, services, and more through the Portal section of the Content pane in ArcGIS Pro.
Does this sound like what you are after?
The following shows the button to open the data in desktop. My customer is trying to open this is arcPro, but it reverts to Arc 10.8x. How do I specify that button to open arcPro?
Thanks for the screenshot. It seems like you are using a version of ArcGIS Enterprise 10.7.x, is that correct? I took a look at the Item Details documentation page for ArcGIS Enterprise 10.6.x and 10.7.x, and believe I have found the answer.
In 10.6.x, we get " For example, maps open in Map Viewer, scenes open in Scene Viewer, and layer packages open in ArcGIS Desktop."
In 10.7.x however, we get "For example, maps open in Map Viewer, scenes open in Scene Viewer, and layer packages open in ArcGIS Pro."
Indeed, when I try this out in 10.7.x, I see that I am given the option to open a feature layer in ArcGIS Pro, but not in 10.6.x. So it seems that this was a feature that was added alongside the 10.7.x of ArcGIS Enterprise.