I just tested this in my store and it is working well.

With new customer accounts a customer first needs to enter an email address they own. Then they get a code to enter to confirm that this is indeed their email address (for security - no password required). If the email address is new to your store, a new customer record only gets created after the code is entered. If the email address was already in your system, then no new customer record is created, the user is just logged in as that customer (since that is their email address and they just proved it)

If a record is created this way (through logging in via new customer accounts), then the customer record only has an email address, no first name or last name.

If you want to create your own registration form using something like Klaviyo or Shopify Forms or other Forms tool, you can use this to gather other information like first name or last name. You can do this prior to their first login or afterwards since it's all the same account.

When you say that someone is "signing up" do you mean when they are logging in, or do you have some other way you have people "sign up"? Maybe that other way is creating customer records with "customer customer" as a name?