Something like this should work:

import arcpy

inShapefile = pointsShapefile
checkField = "xyCombine"
updateField = "dplicate"

with arcpy.da.SearchCursor(inShapefile, [checkField]) as rows:
    values = [r[0] for r in rows]

d = {}
for item in set(values):
    if values.count(item) > 1:
        d[item] = 'Y'
    else:
        d[item] = 'N'

with arcpy.da.UpdateCursor(inShapefile, [checkField, updateField]) as rows:
    for row in rows:
        if row[0] in d:
            row[1] = d[row[0]]
            rows.updateRow(row)

Something like this should work:

import arcpy

inShapefile = pointsShapefile
checkField = "xyCombine"
updateField = "dplicate"

with arcpy.da.SearchCursor(inShapefile, [checkField]) as rows:
    values = [r[0] for r in rows]

d = {}
for item in set(values):
    if values.count(item) > 1:
        d[item] = 'Y'
    else:
        d[item] = 'N'

with arcpy.da.UpdateCursor(inShapefile, [checkField, updateField]) as rows:
    for row in rows:
        if row[0] in d:
            row[1] = d[row[0]]
            rows.updateRow(row)

And as @mr.adam suggested, the dictionary is not needed. here is the cleaner version:

import arcpy

def findDupes(inShapefile, checkField, updateField):
    with arcpy.da.SearchCursor(inShapefile, [checkField]) as rows:
        values = [r[0] for r in rows]

    with arcpy.da.UpdateCursor(inShapefile, [checkField, updateField]) as rows:
        for row in rows:
            if values.count(row[0]) > 1:
                row[1] = 'Y'
            else:
                row[1] = 'N'
            rows.updateRow(row)
            
if __name__ == '__main__':
    fc = r'C:\TEMP\crm_test.gdb\test'
    fld = 'Project_Manager'
    up = 'duplicates'

    findDupes(fc, fld, up)

Something like this should work:

import arcpy

inShapefile = pointsShapefile
checkField = "xyCombine"
updateField = "dplicate"

with arcpy.da.SearchCursor(inShapefile, [checkField]) as rows:
    values = r[0] for r in rows]

d = {}
for item in set(values):
    if values.count(item) > 1:
        d[item] = 'Y'
    else:
        d[item] = 'N'

with arcpy.da.UpdateCursor(inShapefile, [checkField, updateField]) as rows:
    for row in rows:
        if row[0] in d:
            row[1] = d[row[0]]
            rows.updateRow(row)

Something like this should work:

import arcpy

inShapefile = pointsShapefile
checkField = "xyCombine"
updateField = "dplicate"

with arcpy.da.SearchCursor(inShapefile, [checkField]) as rows:
    values = [r[0] for r in rows]

d = {}
for item in set(values):
    if values.count(item) > 1:
        d[item] = 'Y'
    else:
        d[item] = 'N'

with arcpy.da.UpdateCursor(inShapefile, [checkField, updateField]) as rows:
    for row in rows:
        if row[0] in d:
            row[1] = d[row[0]]
            rows.updateRow(row)