Check if restarting ArcGIS Pro resolves the issue.
Use Add Field geoprocessing tool in case you are not able to add field from the attribute table.
I am also having this issue when going through Data -> Fields.
Also, I can't find a way to activate the "Add" button through the attribute table:
Get out of an edit session if you are in one. And if the Add doesn't come back, then try Jayanta's suggestion
Pro doesn't have edit sessions like ArcMap. Editing is always active if you have an edit tool selected. On all layers not specially marked as not editable. . .
Teaching Intro to GIS with Pro this year ... for the first time. It can really be a mind bender.
True... I still forget to say... save your edits if there were any
If you have a Basic level license, there's a known issue with adding fields to shapefiles through the UI. Thread about it here. The workaround is to use the Add Field geoprocessing tool. Unfortunately, it looks like the bug is not fixed in Pro 2.1.