A MoneyGram number refers to the reference number used by customers to receive funds that have been sent to them. These numbers are given to the person sending the funds, which they need to provide to the person picking up the money so that they can verify and collect their funds.
This number is not the only type of verification that a customer may need to verify their identity. The MoneyGram locations can request that someone picking up money show an approved type of photo identification, such as a drivers license or a passport. It may also require something to verify the customer's address, like a utility bill or a bank statement.