I don't think there's a pre-built tool to do this. If it doesn't matter where in the polygon the point is moved it should be a possible using a fairly short script by replacing the point geometry by the centroid geometry from the polygon. Something like this should work (not tested):
import arcpy
# starting data
point_featureclass = "path_to_point_featureclass"
polygon_featureclass = "path_to_point_featureclass"
# compose a dictionary with the the ID and new geometry as pairs
new_point_dict= {row[0]: row[1] for row in arcpy.da.SearchCursor(polygon_featureclass , ["ID_FIELD", "SHAPE@XY"])}
# overwrite point geometry with centroids from point feature class
with arcpy.da.UpdateCursor(point_featureclass, ["ID_FIELD", "SHAPE@"]) as cursor:
for row in cursor:
if row[0] in new_point_dict:
row[1] = new_point_dict[row[0]]
cursor.updateRow(row)