This is how I would test and investigate the problem :-
I find that via Options / Cookies I have two columns, Cookies to Delete and Cookies to Keep.
Under Cookies to Keep I have "google.co.uk", and when selected I see a Firefox icon on the bottom of the displaying which denotes that it is a normal Firefox cookie.
After running CCleaner to clean the system, the Cookies to Delete column is empty because they have been deleted,
and "google.co.uk" is still present under Cookies to Keep which proves that either it was not deleted,
OR that CCleaner is still remembering what was there because of a failure to update to the latest reality.
I again select "google.co.uk" and click the centre column arrow to move this to the other column,
and if successful it proves that I am looking at reality and not a memory.
Then of course I select the cookie and put it back to into "Cookies to Keep".
If all Google cookies are kept, then it must be something else, maybe a cookie that does not look like Google.
Assuming that CCleaner has run and Analyze confirms that there are neither any files nor cookies to delete,
Then I would go to the Google site and use "the 2 step verification process" and then exit the browser as normal.
Then I would launch CCleaner and ANALYZE but not run through its cleaning actions.
Hopefully I will then find under "Cookies to Delete" some new cookies,
some of which may be something special that are created during the 2 step verification process, and moving these to "Cookies to Keep" will solve the problem,
otherwise Analyze might find some TEMP or LOG files or some other recent creation that needs to be excluded from removal.