Below are the available options I can find in the settings:
How can I pick a specific projection such as Web-Mercator (EPSG:3857)?
For Web-Mercator Decimal Degrees is generally used. If you have a layer in Web-Mercator and you just want to see units in Web-Mercator you can choose decimal degrees from this drop down menu. Or Just open a new ArcMap and see the units will be Unknown. Add the layer which is in Mercator projection and the units will be updated to show decimal degrees if the layer is in Mercator.
If you have a data on different coordinate system and you want it to be displayed in Web Mercator you can search for On the fly projection. So For a new ArcMap instance (or new dataframe) the projection depends on the first layer. Which ever coordinate system is for first layer the respective units gets displayed. If new layers are added of other projection system they are projected on the fly. So if you have an ArcMap Instance open with layers in other projection system you can add this Web-Mercator layer and apply the transformation (if required) and change the unit to Web-Mercator .
You will have to set the data frame's coordinate system to EPSG::3857. Open the data frame's property page by right-clicking in the map (or Layers in table of contents) and select Properties.
Select the Coordinate System tab. In the search box, type 3857 and okay the dialog.
Depending on what's already occurred, the status bar should now display values in meters. If it doesn't, and is showing degree values instead, open the data frame properties again.
Select the General tab. Change the Display Units to be meters.