GoDaddy customer number is your personal identification number at GoDaddy. It has several function, namely it can be used for retrieving your password, identifying yourself at phone support, accepting domains to your account and many more.
Finding your customer number is really straightforward, since it is shown at two different places right after login (see below).
It is advisable to record this number in your notebook. As soon as you cannot login to your dashboard for any reason (e.g. forgotten password or hacked account) you cannot use this screen anymore and will be in trouble when trying to take back the control again without the number.
GoDaddy customer number is also important when a domain is sold to another GoDaddy customer. The seller needs the customer number of the buyer to be able to push the domain to the buyer’s account.
The customer number is not a public information and it is not recommended to share it without reason. However, the security risk is limited even if the number is in bad hands – customer number alone is not enough to change or retrieve passwords. If the answers of your security questions are not trivial and you are the only one who can check your emails than there is no reason to worry.