Perhaps run this calculator expression on field Shape of your target:

def getSegment(ID):
 f = arcpy.AddFieldDelimiters("HULLS","LABEL")
 q="%s='%s'" %(f,ID)
 with arcpy.da.SearchCursor("HULLS", 'Shape@',q) as cursor:
  for row in cursor:  pair=row[0]
 with arcpy.da.SearchCursor("TARGET", 'Shape@',q) as cursor:
  for row in cursor:  shp=row[0]
 m=shp.difference (pair)
 return m
#---------------
getSegment( !Label! )

BEFORE:

AFTER:

Perhaps run this calculator expression on field Shape of your target:

def getSegment(ID):
 f = arcpy.AddFieldDelimiters("HULLS","LABEL")
 q="%s='%s'" %(f,ID)
 with arcpy.da.SearchCursor("HULLS", 'Shape@',q) as cursor:
  for row in cursor:  pair=row[0]
 with arcpy.da.SearchCursor("TARGET", 'Shape@',q) as cursor:
  for row in cursor:  shp=row[0]
 m=shp.difference (pair)
 return m
#---------------
getSegment( !Label! )

BEFORE:

AFTER:

Python expression above assumes:

• your target layer is stored as shapefile and named "TARGET" in the table of current mxd.
• eraser layer stored as shapefile and named "HULLS" in the table of current mxd.
• Common field in both layers is text field called "Label". If this not a case create such field in both and populate them from existing common fields.

NOTE: script modifies shapes in existing target layer, so create backup copy first.

First learn how to use advanced field calculator in arcgis.