This is a configuration setting and can be changed per envelope (as well as the Template), This is an Advanced Option setting called "Allow recipients to change signing responsibility". This also can be disabled at an Account level by your DocuSign Admin in Signing Settings.

I thought about that, but don't you have to actually do something on the document to reassign the responsibility and shouldn't it then be signed by that new person with their name? In this case, the person who had the email forwarded to them signs it and the name still shows up as the original recipient.

My fear is they are "forwarding" the email to another person and not using the "Assign to Someone Else" functionality. I can guarantee that if a Recipient chooses the "Assign to Someone Else" they add a Name and Email address of the new Recipient who then gets a notification to sign and they should be signing their name. However if the Recipient "forwards" the email, which they should never do, then all bets are off as to the expected outcome.

There is also documentation with the best practices and other solutions here https://support.docusign.com/articles/Forwarded-Signing-Invitation-Emails. Please note, the process of forwarding a DocuSign email to someone else is not recommended for a series of reasons.

How can we send completed envelopes to be signed again by other recipients using rest api