ArcPy UpdateCursor and Dictionary

You have some problems in the update for. 1) I don't understand why you want to iterate the dictionary, you already have the key in row[0]. 2) This comparison v == min(v) it is always false v is an array and min(v) is an interger. 3) Related with 2, you are only updating on true case.

So this is a fix to your code,

# Start_End_UpdateAttribute.py
# purpose: For each unique 'Id' value, populate the 'Start_End' field in the valley centerline points shapefile attribute table
# with "start" and "end" using the value of the 'OID_copy' field as criteria

import arcpy
import traceback

arcpy.env.workspace = r'D:/GIS_data/Regional/Washington/ValleyBottom_UC_SF/ValleyBottom_UC_SF_centerlines/Wenatchee'
fc = r'D:/GIS_data/Regional/Washington/ValleyBottom_UC_SF/ValleyBottom_UC_SF_centerlines/Wenatchee' \
     r'/VC_ChiwawaRiver_split_verts_Copy.shp'
fields = ['Id', 'OID_copy', 'Start_End']
idDict = {}
cursor = arcpy.da.SearchCursor(fc, fields)

try:
    for row in cursor:
        id = row[0]
        oid = row[1]
        stend = row[2]
        idDict.setdefault(id, []).append(oid)
except:
    print 'An error occurred'
    traceback.print_exc()
del cursor
print idDict  # output is {key: [value, value], key: [value, value], etc}

cursor2 = arcpy.da.UpdateCursor(fc, fields)

try:
    for row in cursor2:
        v = idDict[row[0]]
        if row[1] == min(v):
            row[2] = "Start"
        else:
            row[2] = "End"
        cursor2.updateRow(row)
    print 'Updated \'Start_End\' field'
except:
    print 'An error occurred - part 2'
    traceback.print_exc()
del cursor2

You can use pandas library. If your table is sorted by Id and OID_copy you dont need to use a dictionary:

import arcpy
import pandas as pd

fc = r'C:\GIS\data.gdb\points'
fields = ['Id','Start_End']
df = pd.DataFrame.from_records(data=arcpy.da.SearchCursor(fc,fields), columns=fields) #Create pandas dataframe using da.SearchCursor

l = df.groupby(fields[0]).cumcount().tolist() #https://stackoverflow.com/questions/23435270/how-to-add-sequential-counter-column-on-groups-using-pandas-groupby
l = ['Start' if x%2==0 else 'End' for x in l] #If even cumcount number 'Start' else 'End'

with arcpy.da.UpdateCursor(fc, fields[-1]) as cursor:
    for row, startend in zip(cursor, l):
        row[0] = startend
        cursor.updateRow(row)