How to count how many keys are duplicated in global?

Globals can't have duplicate keys.

After I execute your code and call

zw ^Data 

I get this output:

^Data("Athens")=7
^Data("Boston")=3
^Data("Cambridge")=1
^Data("London")=4
^Data("New York")=2

Did you try to set it as is?

There is no way to get duplication as you describe. 

You can check If there is already the same value with $DATA function.

Write $Data(^Data("Athens"))

Where have you seen such a global?

Keys in global are unique. If you set the value of a global for the same key you'll override the value for the key.

Like @Eduard Lebedyuk wrote if you run 3 commands:

Set ^Data("Athens") = "5"
 Set ^Data("Athens") = "6"
 Set ^Data("Athens") = "7"

You'll first have 5, then override it with 6 and end up with value 7 for the key "Athens".

to count the number of changes you may use this approach

 Set ^Data("Cambridge") = "1" if  $increment(change("Cambridge"))
 Set ^Data("New York") = "2" if  $increment(change("New York"))
 Set ^Data("Boston") = "3" if  $increment(change("Boston"))
 Set ^Data("London") = "4" if  $increment(change("London"))
 Set ^Data("Athens") = "5" if  $increment(change("Athens"))
 Set ^Data("Athens") = "6" if  $increment(change("Athens"))
 Set ^Data("Athens") = "7" if  $increment(change("Athens"))

zw change

change("Athens")=3
change("Boston")=1
change("Cambridge")=1 
change("London")=1 
change("New York")=1

Global keys are always unique! I think you are confused because in a traditional sql environment you can only insert a row using a key and value pair once. After that, to change the value you do an update!

In cache, there is no distinction between setting a global node the first, the second or the third time.

insert into student(studentid, name) values(1, 'john')

to change this, you must do an update:

update student set name = 'john smith' where studentid  = 1

in cache, you can simply do this:

set ^student(1)="john"

and then do:

set ^student(1)="john smith"

There is no obvious distinction between n insert and an update!