They write it down.
It's in a database . . . there's no "remembering" involved.
Your tax identification number is also known as your social security number that will be found on your government issued social security card. If you do not have a card or don't remember your number, you can apply for a replacement card with your local Social Security office.
Social security number
Social security number
Social security number
Social security number
Social security number
Social security number
Social security number
Social security number
Social security number
As you ask, it is not possible if someone try to fing general background can find Social Security number. Only Social Security Authority or Government Authority can find it if it is needed or you requested it.
The federal government is responsible for collecting and administering Social Security. State governments do not collect Social Security taxes or administer the program.