Yeah this is certainly a confusing one and looks like maybe an accident in ArcGIS Online development or maybe a hint at what we'll see in an upcoming release.
I think the confusion here is that you can add markup layers to Map Viewer via the 'Content' tab:
But if you open up a web map in Map Viewer, you can't browse and find markup layers to add in... It can only be done via the method in the screenshot above ^
And as mentioned by @RM5 , the item Overview only shows the option to 'Open in Map Viewer Classic'...
I'm going to re-open the defect record for this and will make mention of it here.