The more information agents have about the customer, the better they can help them. A customer form can be used to capture more information about them - have they contacted support before, are they paying customers, what were the features promised to them and when.
This form doesn’t necessarily have to be filled out by customers. After getting basic information from the customer like their email address, the remaining information can be filled out by the agents who talk to these customers. The next time the customer contacts support, the agent dealing with the customer can go through the information in the form and get context.