In ArcMap there was a way to remove individual labels. Is there a similar functionality in ArcGIS Pro? For example I labeled some streets from a shapfile and would like to delete some unnecessary road names. I would usually use Adobe Acrobat DC to edit this but I do not have access to this application right now.
In ArcMap, there were two types of annotation - map annotation and geodatabase annotation. Map annotation could be created by converting labels to annotation with the radio button "stored in the map" clicked. Another way to create map annotation was via the Draw Toolbar in ArcMap. Map annotation as the name implies is stored in the map document (i.e. *.mxd). I "think" you were using Map Annotation vs. labels. Labels are managed as a group so all label properties are applied to that label group. One could not delete a single label as a result.
ArcGIS Pro only supports geodatabase annotation and labels. Map annotation is not supported. A workaround is to create/use Map Notes in ArcGIS Pro. You may learn more here.
From the labeling side, you can use SQL queries in your label classes to specify which features to label or not to label.