In the EU, customers have a right to be forgotten, which means that you might have an obligation to erase the customer's information from the order if they ask for it.
This right is based on the General Data Protection Regulation in the EU.
There could be some exceptions to this, e.g., if some other law or regulation requires you to save information for other purposes such as accounting or taxation.
How to anonymize an order
Go to Sales > Orders.
Find and open the order that you need to anonymize.
Read the notification carefully, and if you are sure that you can erase the information, click Anonymize.