Anonymisation is an admin-only function, but we do have customers who frequently automate it into their existing compliance workflow since it can be called via the API.
IANAL, but this seems like an easy case: You can’t authenticate the request so you can’t process it.
This sounds like a process problem that needs to be fixed - people’s names are not unique so a request coming down the pipeline to you saying “Delete Jane Doe’s account” is invalid. What would your company do if Gavin Hudson wrote in asking to delete his account? There needs to be some unique identifier associated with the request trackable across your systems, be that email address or an internal account ID.
If the unique identifier is “J.doe@business.de” then your response ought to be “no account exists with that email address”.